What was the first example of a Nazi-based civilization in science fiction, either in action or appearance? Some sort of proof that the inspiration was intentional would be be preferable, but I'll accept an answer if you can make a strong case.
Actual Nazis, such as the ones Captain America fought, don't count.
1933: "The Robot Technocrat" by Nat Schachner appeared in Wonder Stories, March 1933, apparently never reprinted. (Edit. That issue of Wonder Stories has been scanned into the Internet Archive; thanks to lucasbachmann for pointing this out.) I don't have a copy, so I'm going by Everett F. Bleiler's review in Science-Fiction: The Gernsback Years, quoted below.
In this world of 1954, the Depression still holds sway and the United States is politically shattered intoa welter of political parties: Extreme Communist, Communist, Socialist, Social-Laborite, Middle Class, Patriots, Fascist, Monarchy, Aristocrat, etc. Each of them constitutes a small dictatorship, with absolute leader, private army of storm troopers or strong-arm men, and a policy of ruthlessness. Most troublesome of all is the Nationalist Party headed by Adolph Hiller (whose referent the reader will not need a biographical dictionary to identify). The most palatable is the Reconstructionist movement headed by Corbin, who is a humane, reasonable man who does not see force as his first option. Europe and Asia have all gone either Communist or Fascist.
Corbin has received a summons from the long-disappeared Professor Kalmikoff, and together with his military leader General Wingdale and an attaché visits the scientist's hidden laboratory.
Kalmikoff explains. Since his withdrawal from society he has secretly been working on a supercomputer (my term) for the understanding and prediction of future history. By analyzing twenty variables, including those taken from individuals concerned, the machine will indicate what will happen. In a sense, as the scientist explains, it is a perfected, more sophisticated version of what the earlier Technocrats tried to do in simplistic fashion.
Corbin and Kalmikoff decide on a rash action: Kidnap all the party leaders, bring them to the lab, show them the machine, and let the machine predict the future that each man will create if he attains power. The caper succeeds, with one exception, Hiller is not picked up. But, as the machine is in operation, Hiller appears with his private army and takes over the situation.
The machine has shown disastrous futures for those it has "interviewed," and when Hiller voluntarily enters it, it shows a reign of terror. Hiller thereupon starts to execute everyone, but is interrupted in a grand shoot-out by friendly troops.
Hiller and a couple of the other extremists are killed. The survivors, now somewhat shocked into reason, agree to support Corbin when the machine predicts a favorable future under his leadership.
I see there have already been some good answers, but I have another candidate to offer.
Back in the 1930s, there was a pulp magazine series called Operator #5 (soon switched to Secret Service Operator #5). The series hero was a brave young secret agent (working for the USA), Jimmy Christopher, who was a master of disguise and had some fictitious high-tech gadgets which he habitually carried on his person for sudden emergencies. In those days, the typical "pulp novel" was a self-contained story, with the hero having decisively defeated the bad guy (or a whole collection of bad guys) by the end of that same tale -- the obligatory happy ending, you know.
But Operator #5 went through a lengthy ongoing storyline that is sometimes called "the War and Peace of the pulps." Written by Emile C. Tepperman (though he was working under the magazine's house name of "Curtis Steele," as did anyone else who wrote stories about Operator #5). This saga filled up 13 consecutive issues of the magazine, with about 750,000 words of prose when all was said and done, and the whole thing is now sometimes called "The Purple Invasion."
It began with Operator #5 #26 (June 1936), with the long main story titled: "Death's Ragged Army." (There were some unrelated shorter stories toward the back of the issue.) This was the beginning of a saga of an imaginary war which was only resolved in #38 (March 1938), titled: "The Siege That Brought the Black Death."
I own an e-book version of that first installment. It starts with a scary premise: As the opening chapter sets the stage for us, it becomes apparent that the legions of "the Central Empire" have already invaded and easily overrun the northeastern corner of the USA. (The New England states and the state of New York, and maybe a few other bits and pieces in that region.) And they show no signs of stopping there. These legions of soldiers are led by senior officers who often have Germanic-sounding names -- examples include "Colonel Mainz" and "Baron Oscar zu Lambrecht."
The Central Empire has advanced weapons which Operator #5 recognizes are far more powerful than anything the United States military possesses. This includes a mysterious "Green Gas" which, we are assured, could be used to sterilize the entire USA in a matter of weeks if the ruler of the Central Empire ("Emperor Maximilian") were so bloodthirsty as to feel the need to do so. Fortunately, he prefers to conquer as much of the American population as possible so that they can keep working hard and contributing to his empire's economy. Later in the series, the Central Empire will use germ warfare as well. (And they very nearly overrun the entire USA before the tide finally turns in favor of Operator #5 and his fellow Americans as they fight their way to victory.)
By the end of the first installment, Maximilian has died, but this is not good news for the Americans. The old Emperor's son, Rudolph, has inherited control of the Central Empire, and he is known to be much more cruel and bloodthirsty than his daddy ever was. (Any resemblance between the names "Adolf" and "Rudolph" is, of course, not coincidental.)
So if you count "alternate history of a scary possible future" and "the enemy uses advanced weapons that didn't exist at the time the story was written" as being strong symptoms of science fiction, then this qualifies as a very early science-fictional treatment of a militaristic group who were never called "Nazis" or "Germans" in the texts of the stories, but who were obviously based on what Adolf Hitler was doing in Central Europe to make Germany become once again a military force to be reckoned with. (And Hitler wasn't exactly shy about sharing his desire to grab more and more real estate to his collection, either, although these stories started being written before he'd managed to annex Austria, for instance, and before he actually declared war on anyone.)
A bit ambiguous, but Olaf Stapledon's novel Star Maker (1937) showed a variety of alien worlds that were all going through a similar "crisis" of civilization, with some responding in ways that were clearly inspired by fascists, and probably by Nazis specifically. This article has a good summary of how Stapledon depicted this "crisis", which Stapledon saw as having a lot to do with the loss of traditional religious belief in a "Love-God" along with the potential for greater destruction made possible by technology. On one particular world encountered early on in the story, an alien species the narrator calls the "Other Men" is seen to be in this type of crisis period, and one of the technologies which is causing problems on this world is a sort of virtual reality experience that worked through "direct stimulation of the appropriate brain centers" creating experiences that involve "touch, taste, odor, and sound" (apparently not sight, but these aliens were said to have more well-developed senses of scent and taste than hearing or sight). They then developed a system by which a being could "retire to bed for life and spend all his time receiving radio programs", their bodies being fed and exercised automatically, and one of the reactions to this new type of technology was a political movement that is specifically referred to as an analogue of fascism, and the discussion of "racial hatred" probably was meant to invoke Nazism specifically (since Mussolini's fascism didn't place much emphasis on race, at least not in its earlier years before the wartime alliance with Hitler):
Many of the great economic masters, though they had originally favored radio-bliss in moderation as an opiate for the discontented workers, now turned against it. Their craving was for power; and for power they needed slaves whose labor they could command for their great industrial ventures. They therefore developed an instrument which was at once an opiate and a spur. By every method of propaganda they sought to rouse the passions of nationalism and racial hatred. They created, in fact, the "Other Fascism", complete with lies, with mystical cult of race and state, with scorn of reason, with praise of brutal mastery, with appeal at once to the vilest and to the generous motives of the deluded young.
The 1936 book War With the Newts by Karel Čapek (who invented the term 'robot' in his play R.U.R.) could be another candidate, with the newts trying to break up human land masses to provide more "living space" for themselves, in a way similar to Hitler justifying Nazi invasions of other countries in terms of the German people needing more Lebensraum, commonly translated as "living space". The wikipedia article references some sources that suspect the story was inspired at least in part by Nazism:
By the early 1930s, the author's country of Czechoslovakia was in a precarious political situation. Čapek became concerned by the developments of National Socialism in Germany and the rise of the Soviet states to the east. He began writing his Apocryphal Tales, short allegorical pieces that picked up on the anxiety felt by many Czechoslovakians at the time. These may have provided the impetus for War with the Newts, which was written over four months in the summer of 1935.[here the wikipedia article references the book Karel Čapek: Life and Work by Ivan Klima]
and also
Darko Suvin has described War with the Newts as "the pioneer of all anti-fascist and anti-militarist SF".[here it references Suvin's entry in the book Twentieth-Century Science-Fiction Writers]
However, this article argues that the book's satire has a somewhat different target:
Throughout the book, underneath the stylistic tricks (typeface switches, footnotes, people who speak in newspaper headlines, fake academic articles), the real subject of Čapek’s scorn is modern commerce and capitalism.
And it also notes that
the book is broad enough to be open to numerous, and even contradictory, interpretations. (See here [PDF link], where the editor of Penguin’s Central European Classics series, emphasises the equally powerful impression of the Newts representing the Nazis.)
According to wikipedia's article on Space Nazis, Edgar Rice Burroughs (of Tarzan fame) satirised the Nazi Party by situating a fascist political faction called "the Zani" on the planet Venus in his 1938 serial 'Carson of Venus'.
During our visit with Taman and Jahara we learned many things concerning them and Korva. Following a disastrous war, in which the resources of the nation had been depleted, a strange cult had arisen conceived and led by a common soldier named Mephis. He had usurped all the powers of government, seized Amlot, the capital, and subdued the principal cities of Korva with the exception of Sanara, to which many of the nobility had flocked with their loyal retainers. Mephis had imprisoned Jahara's father, Kord, hereditary jong of Korva, because he would not accede to the demand of the Zanis and rule as a figurehead dominated by Mephis. Recently rumors had reached Sanara that Kord had been assassinated, that Mephis would offer the jongship to some member of the royal family, that he would assume the title himself; but no one really knew anything about it.
In this instance, Mephis is clearly a stand-in for Hitler and the Zanis are basically the Nazi Party in Spaaaaaaace!
An honourable mention should go to 'Rocket Ship Galileo' by Robert Heinlein from 1947 as one of the first mentions of a Nazi colony outside of Earth.
As they set up a radio to communicate with the Earth they pick up a local transmission, the sender of which promises to meet them. Instead, their ship is bombed. Fortunately, they are able to hole up undetected in their hut and succeed in ambushing the other ship when it lands, capturing the pilot. They discover that there is a Nazi base on the Moon. They bomb it from their captured ship and land. One survivor is found, revived, and questioned.
One possibility is E.E. Smith's Skylark Three (1930) with the Fenachrone, a nonhuman species with their own planetary civilization who consider themselves "supermen" and plan to exterminate all inferior species (i.e. everybody) in the galaxy.
The Nazis were not yet in power in 1930 but were making a big noise in German politics. So they might have inspired or influenced the Fenachrone.
Not as early as some of the answers given (are they all science fiction Nazis?) but I want to mention "Patterns of Force", the Star Trek TOS episode from 1968.
We had an entire planet with swastikas and jackboots etc
Echoing M.A. Golding, we will boldly suggest the existence of Nazi civilizations in science fiction as early as 1921 — twelve years prior to the start of the Third Reich, or the origination of Nazi Germany.
Original question did not limit to just American authors, and European fiction writers were all writing science fiction throughout the years. German science fiction was a part of this, doing everything that science fiction does: observing the human nature at the time, including culture and politics of the age.
One such author is Hans Dominik, whom Wikipedia notes as: https://en.m.wikipedia.org/wiki/Hans_Dominik_(writer)
One unsympathetic biographer, William B. Fischer, has written of him that “In many ways he was shallow-minded and ignorant, yet pompously opinionated about literature, science, and politics. He was also a racist and chauvinist whose attitudes and works easily lent themselves to the aims of National Socialism.” Despite this, in the 1980s Dominik was still one of the small number of highly popular German-born science fiction writers.
Throughout his career, Dominik appears to have written several science fiction works premised on a Nazi civilization.
His Die Macht Der Drei (The Power of Three) published in 1921:
…is set in 1955. The British Empire threatens to collapse and the conflict with the United States comes to a head. Ultimately, after armed provocations, the British declare war on the USA. At this moment the power of three intervenes: the German Sylvester Bursfeld, the Swede Erik Truwor and the Indian Soma Atma. They have developed a telenergetic radiator that gives them tremendous power. The three men succeeded in releasing bound energy in free space, solving the problem of telenergetic concentration and thereby achieving tremendous effects. The weal and woe of the earth are now in the hands of these three people.
The novel presents the Three as benevolent leaders seeking peace for the good of all nations. The opponent of the three is a charismatic American with hypnotic skills giving him a supernatural ability to rally those to his cause, and uses reprehensible methods as mass extermination to try to become a totalitarian dictator. Everything seems lost until the Indian brings in his ancient culture and wisdom. He is also the only one of the triumvirate to survive.
So in this case, in 1921 German science fiction authored by someone at least one described as a racist chauvinist, the Americans some 34 years in the future were the Nazi civilization run by Literal Adolph Hitler, and the benevolent German socialist was seeking world peace.
