In E.E. Smith's The Skylark of Space Chapter VII "The Trial Voyage" there is the following discussion of the spaceship's possible speed.
To hold down the power-plant, so that the bar won't tear through the ship when we cut her loose," replied Seaton. "Have you any idea how fast this bird can fly?"
"Well, I have heard you speak of traveling with the velocity of light, but that is overdrawn, isn't it?"
"Not very much. Our figures show that with this four-hundred-pound bar"—pointing to the copper cylinder in the exact center of the inner sphere—"we could develop not only the velocity of light, but an acceleration equal to that velocity, were it not for the increase in mass at high velocities, as shown by Einstein and others. We can't go very fast near the earth, of course, as the friction of the air would melt the whole works in a few minutes. Until we get out of the atmosphere our speed will be limited by the ability of steel to withstand melting by the friction of the air to somewhere in the neighborhood of four or five thousand miles per hour, but out in space we can develop any speed we wish, up to that of light as a limit."
"I studied physics a little in my youth. Wouldn't the mere force of such an acceleration as you mention flatten you on the floor and hold you there? And any sudden jar would certainly kill you."
So the characters are certainly aware that Einstein had calculated that mass would increase with velocity so that it would be impossible to reach the speed of light.
In Chapter VIII "Indirect action":
Regaining his self-possession as the wisdom of his friend's advice came home to him, Seaton sat down and pulled out his pipe. There was a tense silence for an instant. Then he leaped to his feet and darted into his room, returning with an object-compass whose needle pointed upward.
"DuQuesne did it," he cried exultantly. "This baby is still looking right at him. Now let's go—make it snappy!"
"Not yet. We should find out how far away they are; that may give us an idea."
Suiting action to word, he took up his stopwatch and set the needle swinging. They watched it with strained faces as second after second went by and it still continued to swing. When it had come to rest Crane read his watch and made a rapid calculation.
"About three hundred and fifty million miles," he stated. "Clear out of our solar system already, and from the distance covered he must have had a constant acceleration so as to approximate the velocity of light, and he is still going with full...."
"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."
"That's right. Another good theory gone to pot. But how do you account for his distance? D'you suppose he's lost control?"
Smith first wrote The Skylark of Space between 1915 and 1921. The quotations are from the Project Guttenburg text.
This etext was produced from Amazing Stories August, September and October 1928. Extensive research did not uncover any evidence that the U.S. copyright on this publication was renewed.
Other notes and a list of corrections made will be found at the end of the book.
So any earlier science fiction mention of the Speed of light or of Einstein's theory of relativity stating that the speed of light was the absolute speed limit will have to be published before The Skylark of Space in 1928.
David W.'s answer has a mention of the speed of light, but not as any sort of absolute speed limit, from 1882.
So the first mention of the speed of light in science fiction was in 1882 or earlier, while the first mention of the speed of light as an absolute speed limit would be sometime between 1882 and 1928.
The Skylark of Space was one of the earliest stories of interstellar travel but not the very first. So there are several possibilities to check for earlier mentions of the speed of light as a cosmic speed limit.