I have just watched Space Battleship Yamato (2010) and they used the term "warp" to denote a type of drive. I think Star Trek is the obvious place where this term was first used to describe faster than light travel, is it where it originated?
I found a reference to an "an interstellar space-warp drive" in Future science fiction: Volume 1, Issue 1, edited by Robert W. Lowndes. That was in was either "Nobody Saw the Ship" by Murray Leinster (my guess) or "The Miniature Menace" by Frank Belknap Long. The publication date was May 1950.
Per aneroid's suggestion, I had first done a Google ngrams search to get to this point. If there's an earlier reference to "warp drive" or "Warp drive" (entirely possible), then it hasn't yet been scanned by Google books.
Edit to add: For a more famous story, check out this excerpt from "I, Robot" (by Asimov), also published in 1950:
"You get it, chief?" The general manager was wildly jubilant. "You get it? There isn't any industrial research group of any size that isn't trying to develop a space-warp engine, and Consolidate and U.S. Robots have the lead on the field with our super robot-brains." (p. 145)
(This one was found by searching for "space warp" instead of "warp drive".)
Further edit to add: Clearly the phrase "space warp" was already in use by 1947, Google ngrams notwithstanding, as these notes from a symposium suggest:
The term "space warp" does not mean anything without elaborate explanation.
For an earlier use of FTL travel that does not use the word warp, there's Gray Lensman, written in 1939 by E. E. Smith. He refers to a "5th order drive" that can "voyage anywhere in the universe at millions of times the speed of light."
Edited to add: With thanks to @user14111 for pointing it out in the comments, there's an even earlier reference (written between 1915 and 1921) to FTL travel, The Skylark of Space, in which E. E. Smith writes:
Hurtled onward by the inconceivable power of the unleashed copper demon in its center, the Skylark flew through the infinite reaches of interstellar space with an unthinkable, almost incalculable velocity—beside which the velocity of light was as that of a snail to that of a rifle bullet; a velocity augmented every second by a quantity almost double that of light itself.
I haven't read the whole story yet, but it's not clear to me if E. E. Smith was aware of Einstein's special or general theories of relativity.
warp drive (n.) 1948
As far as the Historical Dictionary of Science Fiction knows, the earliest citation for "warp drive" is in the 1948 novella "Flight of the Starling" by Chester S. Geier in Amazing Stories, January 1948, available at the Internet Archive. Quoting from p. 10, col. 2:
The atomic engines of ordinary space vessels attained only a small fraction of the inconceivable velocity which Alward claimed for the warp-drive of the Starling.
warp (v.) 1946
According to the Historical Dictionary of Science Fiction, the earliest known use of "warp" as a verb meaning "to travel through space by way of a space warp" is in the short story "Placet is a Crazy Place" by Fredric Brown in Astounding Science Fiction, May 1946, available at the Internet Archive. Quoting from p. 129, col. 1
Tomorrow the Ark would leave Earth, with the shipment of conditioner that would solve one of our problems—and with whomever Earth Center was sending to take my place. It would warp through space to a point a safe distance outside the Argyle I-II system and come in on rocket power from there. It would be here Friday, and I'd go back with it. But I tried not to think about that.
space-warp generator (n.) 1944
Among the Historical Dictionary of Science Fiction's citations for "space warp" I found this one from the novelette "Circle of Confusion" by George O. Smith (as "Wesley Long") in Astounding Science Fiction, March 1944, available at the Internet Archive. Quoting from p. 53, col. 2:
The alphatron is still in fine shape, and the space-warp generator can still do a job.
warp (v.) 1938
The word "warp" is used in reference to space travel in the 1938 novelette "Men Against the Stars" by Manly Wade Wellman, first published in Astounding Science-Fiction, June 1938, available at the Internet Archive. Quoting from p. 8, col. 2:
"Sixty ships, Tallentyre. Sixty of 'em—and two hundred and forty-two men started from Earth. Fifty-six ships, and two hundred and twenty-two men reached Luna Port. Eighteen men lost on that little hop. Four ships blew their tubes—and that bloody six-man experiment first of all.
"But fifty-six ships landed, and we warped 'em off to Mars. And how many of those fifty-six got through?" His grating scream roared in the cubbyhole office and pounded through its flimsy metal door. Tallentyre's eyes moved toward the door.
DeWitt's roar dropped to a whisper as the man leaned abruptly forward, close to Tallentyre's moveless, sun-blackened face. "Four. Four got to Mars, my friend. The rest were pretty, red firecrackers in space."
Unfortunately, I don't think this is the kind of "warp" we're looking for. Wellman's spaceships did not have spacewarp drives; they were rocket ships fueled with atomic hydrogen. I don't know what Wellman meant by "warp"; maybe he used it in the sense of "cast, throw, fling".
warp (v.) 1932
Quoting from p. 342, col. 1:
"It is a matter of the proper utilization of two forces, electrical and gravitational," proudly explained Dr. White. "Those two forces, properly used, warp the third-dimensional into the fourth. A reverse process is used to return the object to the third. The principle of the machine is—"
The old man was about to launch into a lengthy discussion, but Henry interrupted him. A glance at his watch had shown him press time was drawing perilously close.
"Just a second," he said. "You propose to warp a third-dimensional being into a fourth dimension. How can a third-dimensional thing exist there? You said a short time ago that only a specified dimension could exist on one single plane."
From p. 342, col. 2:
The light did not waver or sparkle. It did not glow. It seemed hard and brittle, like straight bars of force. The newspaperman, gazing with awe upon it, felt that terrific force was there. What had the old man said? Warp a third-dimensional being into another dimension? That would take force!
From p. 344, col. 2:
In a line stood the men who were to fling themselves into the light to be warped into another dimension, there to seek out and fight an unknown enemy. The line was headed by a tall man with hands like hams, with a weather-beaten face and a wild mop of hair. Behind him stood a belligerent little cockney. Henry Woods stood fifth in line. They were a motley lot, adventurers every one of them, and some were obviously afraid as they stood before that column of light, with only a few seconds of the third dimension left to them. They had answered a weird advertisement, and had but a limited idea of what they were about to do. Grimly, though, they accepted it as a job, a bizarre job, but a job. They faced it as they had faced other equally dangerous, but less unusual, jobs.
From p. 345, col. 2:
Then he knew. He was not alone. Here, in this one body were the bodies, the brains, the power, the spirit, of those other ninety-eight men. In the fourth dimension, all the millions of third-dimensional things were one. Perhaps that particular portion of the third dimension called the Earth had sprung from, or degenerated from, one single unit of a dissolving, worn-out fourth dimension. The third dimension, warped back to a higher plane, was automatically obeying the mystic laws of evolution by reforming in the shape of that old ancestor, unimaginably removed in time from the race he had begot. He was no longer Henry Woods, newspaperman; he was an entity that had given birth, in the dim ages when the Earth was born, to a third dimension. Nor was he alone. This body of his was composed of other sons of that ancient entity.
From [p. 346, col. 1[(https://archive.org/details/Astounding_v10n03_1932-06/page/n59/mode/1up):
He felt himself grow, felt his body grow vaster, assume greater proportions, felt new vitality flow through him. It was the other men, the men who were flinging themselves into the column of light in the laboratory to be warped back to this plane, to be incorporated in his body.
Islands of Space by John W. Campbell, Jr., in 1930. (emphasis mine)
"To move around near a heavy mass—in the presence of a strong gravitational field," Arcot said. "A gravitational field tends to warp space in such a way that the velocity of light is lower in its presence. Our drive tries to warp or strain space in the opposite manner. The two would simply cancel each other out and we'd waste a lot of power going nowhere. As a matter of fact, the gravitational field of the sun is so intense that we'll have to go out beyond the orbit of Pluto before we can use the space strain drive effectively."
Also, P. Schuyler Miller in Astounding Science Fiction wrote about this story:
"Arcot, Wade, Morey, and their computer, Fuller, put together a ship which will travel faster than light ... they give us what may have been the first space-warp drive. The concept was simple; to make it plausible wasn't—unless you were John Campbell."
"Every atom of ship load and crew was deflected infinitesimally from the space-time continuum of four dimensions, and thus freed of the ordinary limitations of acceleration and velocity, was driven around space, rather than through it, by a direct reaction against the space warp itself."
First use in movies or first use ever? Geometric warping is used for "basic solutions of the Einstein field equations" - which are based on 'curved space'. Not sure when those solutions were first found but - Einstein: 1879-1955; Riemann: 1826-1866 (~1854); Lorentz: 1853-1928.
"Warping of Wood" as a known fact is probably hundreds or thousands of years old. So I would put 'warp' to denote 'travel' being around Einstein's time.
For something more concrete/definite: Alcubierre drive - that's 1994.
A google books ngrams search might get you its first ever use.
1942: The earliest science-fictional space-warp drive that I know of is in Ralph Milne Farley's novelette "The Immortality of Alan Whidden", first published in Amazing Stories, February 1942 (available at the Internet Archive):
Gravity, Einstein had explained as being merely a warping of space-time in the fifth and higher dimensions, in the vicinity of masses of matter. Alan Whidden himself had supplied the older scientist with the following two-dimensional analogy — Whidden was good at analogies. Stretch out a thin sheet of elastic rubber, Whidden had suggested, and place upon it a number of metal balls of various sizes and weights. Each ball, in proportion to its mass, will distort this two-dimensional rubber sheet into the third dimension, just as gravitating bodies distort our own four dimensional space-time into higher dimensions. The balls on the rubber sheet will roll together, just as masses gravitate together in our familiar space.
The idea now occurred to Whidden: "If I distort the rubber sheet by some other means than the metal balls, as by poking my finger into it, I will have thereby created an artificial gravity. If I move this distortion along, by running my finger along ahead of one of the balls, the ball will follow my finger. Why not then artificially distort space in advance of a space ship, and thereby pull the ship along after the distortion, at any desired speed? It will be like holding a carrot on a pole in front of a mule's nose, to induce him to move."
Whidden took a brief run over to Princeton and discussed the idea with Albert Einstein. The latter disagreed quite emphatically, but could not formulate the reasons for this disagreement. Being a true scientist, Einstein's ideas always developed first as pure hunches, and then required months of abstract research for their formulation, followed by years of observation and experiment for their verification.
But Alan Whidden could not wait for all this. His malady was piling up on him like figures on a taxi-meter. Being the same sort of true scientist as his mentor, he too plunged into a mathematical analysis of his hunch; and when he believe that he had deduced the correct formula for an artificial space-warp, he set about producing it in his laboratory, preferably by electricity.
He found the way. He built a small model space-ship. He was able to cause it to set up a disturbance ahead of it, which would make it gravitate in that direction, its tail held down by a piece of rope. Substituting a spring-balance for the rope, he was able to measure the force of the pull of this artificially induced gravity, and thus learn the law of its strength. Also he found out how to direct the pull in any direction around the little model — this would be useful in steering and stopping his full-sized ship when built.
Unfortunately his full-sized ship is wired slightly differently, with the result that it won't move in space at all, it can only move backwards in time.