Atomic motor in The Skylark of Space
In 1928, E.E. Smith published the novella The Skylark of Space in the periodical Amazing Stories. It features a vessel with an "atomic motor" capable of faster-than-light travel. This seems to be the first working proposal in science fiction for how to power a technology producing faster-than-light travel. There is, however, a caveat regarding the basis for the technology that I will mention below (which is why I include a better candidate — Islands of Space — further down).
Here is a passage from The Skylark of Space in which we see the atomic motor in action:
For forty-eight hours the uncontrolled atomic motor dragged the masterless vessel with its four unconscious passengers through the illimitable reaches of empty space, with an awful and constantly increasing velocity. When only a few traces of copper remained in the power-plant, the acceleration began to decrease and the powerful springs began to restore the floor and the seats to their normal positions. The last particle of copper having been transformed into energy, the speed of the vessel became constant. Apparently motionless to those inside it, it was in reality traversing space with a velocity thousands of times greater than that of light.
The actual technology for achieving this superluminal velocity is never explained. All that is explained is where the power for it comes from. This is the "atomic" in "atomic motor":
"Well, it's the force that exists between the ultimate component parts of matter, if you can understand that. A child ought to. Call in your chief chemist and ask him what would happen if somebody would liberate the intra-atomic energy of one hundred pounds of copper."
The main reason that the operation of the engine is not fleshed out is because the novella relieves itself of this burden by simply asserting that Einstein's theories are incorrect. In other words, there is no obstacle to travelling faster than light, and all one needs is enough power to accelerate to whichever speed one desires.
"But nothing can possibly go that fast, Mart, it's impossible. How about Einstein's theory?"
"That is a theory, this measurement of distance is a fact, as you know from our tests."
Note: Many thanks to @Hypnosifl, @M.A.Golding, @user14111, and @RobertF for their input on this!
Space strain drive in Islands of Space
I believe that the first proposal for a technological mechanism for faster-than-light travel — one that describes its actual design and operation, not only its power source, and which accepts that the speed of light is something to be overcome — appeared in John W. Campbell Jr.'s Islands of Space, a novella first published only three years later (1931), also in Amazing Stories. The vessel in the novella features a "space strain drive", whose description anticipates some facets of Star Trek's famous warp drive.
Notably, this proposed drive also takes into account gravitational effects, in line with Einstein's general theory of relativity.
The novella's lead-in blurb:
As Earth's faster-than-light spaceship hung in the void between galaxies, Arcot, Wade, Morey and Fuller could see below them, like a vast shining horizon, the mass of stars that formed their own island universe.
Later in the novella:
"We might need an aerodynamically stable hull," Wade interjected. "It came in mighty handy on Venus. They're darned useful in emergencies. What do you think, Arcot?"
"I favor the torpedo shape. Okay, now we've got a hull. How about some engines to run it? Let's get those, too. I'll name the general things first; facts and figures can come later.
"First: We must have a powerful mass-energy converter. We could use the cavity radiator and use cosmic rays to warm it, and drive the individual power units that way, or we can have a main electrical power unit and warm them all electrically. Now, which one would be the better?"
Morey frowned. "I think we'd be safer if we didn't depend on any one plant, but had each as separate as possible. I'm for the individual cavity radiators."
"Question," interjected Fuller. "How do these cavity radiators work?"
"They're built like a thermos bottle," Arcot explained. "The inner shell will be of rough relux, which will absorb the heat efficiently, while the outer one will be of polished relux to keep the radiation inside. Between the two we'll run a flow of helium at two tons per square inch pressure to carry the heat to the molecular motion apparatus. The neck of the bottle will contain the atomic generator."
Fuller still looked puzzled. "See here; with this new space strain drive, why do we have to have the molecular drive at all?"
"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."
"I catch," said Fuller. "Now to get back to the generators. I think the power units would be simpler if they were controlled from one electrical power source, and just as reliable. Anyway, the molecular motion power is controlled, of necessity, from a single generator, so if one is apt to go bad, the other is, too."
(Source — extracted from the 1966 Ace paperback edition, which is based on an earlier 1956 version)
Although the above excerpt is from the heavily-revised 1956 version of the novella, the idea of straining space to achieve faster-than-light travel is intact in the original 1931 printing, as seen in this passage:
"Einstein found that the geometry of Euclid was all right here on Earth, where the curvature of space is almost zero, but when he came to consider Mercury, Euclid's geometry was all wrong. Space is curved, and there, near the terrific mass of the sun, it was so badly curved, that Euclid's planes didn't exist, and his geometry was wrong.
"Morey has a new system, and he applied it with amazing results. We have discovered a means for giving any one of the twenty coefficients of space any value we want.
"To do this, we are straining space out of its normal condition, and the usual condition of such strain creeps up. We have to use energy to do it, and in thus straining space, we are storing energy in it. Of course, we have stored energy as space strain for hundreds of years, the induction coil, the condenser and gravity all represent energy stored in space as a strain. The more we follow this mathematical discussion, the more it seems that all energy in the universe is but strain in space: that is, there is no matter, merely energy strain, in space; that there is energy only because space is strained there.
"But, be that as it may, we have a new means of storing energy, and that is more to our point, the means so affects space, that it changes the speed of light, changes the limiting speed of the universe. We can go as fast as we want, for the limit, about us, is changed, by our curving space in a new way.
"The system is fairly obvious. Remember a torus-shaped induction coil encloses all its magnetic field within it, the torus, or 'doughnut' coil, has a perfectly enclosed magnetic field. We built such an enclosed storage coil, and expected to store a few watts of energy in it to see how long it would hold it. We made the mistake of connecting it to the power lines, and it cost us a hundred and fifty dollars at a quarter of a cent per kilo-watt. After that we used the big relux plate electric generating method.
"But that is in the essentials what we have to offer. We give you the job of figuring out the stresses and strains and so forth--we want a ship to visit our neighbors--anywhere within one thousand million light-years."
(Many thanks to @user14111 for typing this out from a copy of the original 1931 periodical! The 1931 version is now available on archive.org here.)
Given the information above, I believe that Islands of Space best meets the requirements of the OP's question.
Regarding the "Bonus" question
As an addendum, if one is interested in faster-than-light travel in science fiction that is not necessarily technological in nature, we have the following.
Astronomer Johannes Kepler wrote a science fiction novel in 1608 called Somnium.
In it, there are demons that can move a human being from one point on Earth to any other possibly instantaneously, and these demons can be called upon to travel great distances whenever necessary.
The main character uses these demons to travel to the moon, for instance. (For some odd reason, travel-by-demon to the moon takes up to 4 hours, whereas travel from any point on Earth to another happens in the blink of an eye. In any case, at least some instances of demon-induced travel in the novel might occur at superluminal speeds.)
From the Wikipedia summary of Somnium:
One day, Fiolxhilde reveals to Duracotus how she learned of the heavens. She tells him about the daemons she can summon. These daemons can move her anywhere on Earth in an instant. If the place is too far away for them to take her, they describe it in great detail.
This précis seems to have been adapted from this passage of the original Latin text:
cujus ope non raro momento temporia in alias oras, quas ipsi dixero, transportor, aut si ab aliquibus longinquitate absterreor, quaerendo de iis tantum proficio, quantum si praesens ibi essem,
which translates as:
Often, in a split second, I was transported by its power to other shores which I selected for myself. If I were kept away from certain places on account of their distance, I gained ground by questioning about those places just as if we were present there.
"Split second" is vague, but Kepler was clearly imagining inhumanly fast travel.
Incidentally, Sagan and Asimov consider this to be the first work of science fiction in general. Since Somnium is the first true work of science fiction, Kepler's demons are also the first faster-than-light mechanism of travel in science fiction — if we assume the usage of "split second" is akin to "instantaneous".
Note: Thanks to @Hypnosifl for referring me to the original text and translation, and thanks to @algiogia for some supporting calculations regarding the demons' terrestrial speed.