As far as I know, the term "space walk" was first used in science fiction (and maybe anywhere) in 1939, in Moon Heaven by Dom Passante:

But that space walk of mine wasn't so very amazing. I've lived here all my life, and like a swimmer who can accustom himself to long periods under water, so have I, by occasional jaunts to the plateau, accustomed myself to void conditions.
(Read more about space walk)

The first description of an untethered (or tethered) spacewalk was probably in Edison's Conquest of Mars (1898), by Garrett P. Serviss, published by New York Evening Journal in 1898.

Update 04-Apr-2024: Here's another version of the untethered spacewalk, this time with a grim reminder of the inexorable nature of Newton's laws of motion. This is from Between Earth and Moon, by Otfrid von Hanstein, published by Wonder Stories Quarterly in 1930:

Quickly he opened the outer door of the airlock. Now he felt just as he had once before, as a student aviator, when he had jumped out of his burning plane at a dizzy height, not knowing whether his parachute would open or not.

Nothing happened. Hesitantly he stepped outside. He had to push himself away from the rocket with his hand he remained floating motionlessly outside.

He hung the iron hook at the end of the rope on a ring on the outside of the rocket and looked about him.

He was unwilling to think; he forced his mind not to take in where he was. He would not remember that the earth lay one hundred and eighty thousand kilometers down below him...

He looked about, and at this moment his vision dimmed and his heart nearly stopped beating...

Now the most frightful thing possible had happened.

Egon was perfectly motionless. The gas container was empty. But the rope which connected him with the rocket was no longer fastened to the latter. Probably the impulse of the writing machine had sufficed to free the hook from the ring. The rope was floating freely in the air, and the rocket was at least twenty meters from the end of it.

(From Between Earth and Moon by Otfrid von Hanstein)

He tried to make swimming motions. Of course it was in vain. He encountered no resistance. He simply lost his equilibrium and had trouble in floating back into it.

The rocket was apparently standing still, but he was moving, to a degree hardly perceptible, more and more away from it. He could not understand how in this moment of the most fearful certainty of death he could think so calmly.

It was quite clear to him what had happened. He, the only person in a diving suit, a hundred and eighty thousand kilometers above the earth, had become a third satellite...

Silently the rocket, the new satellite of the earth, sped on its newly formed orbit. Before it went the tiniest of all bodies in the universe, the grain of dust in space — the living being, still protected from the cold of space by his diving suit, with his lungs still breathing the remnants of the oxygen in the cylinder, his senses still alert, going to meet death with open eyes.

End update.

Update 11-May-2024: I should add that the Manned Maneuvering Unit was anticipated by the self-propulsive space suit described in Bluff of the Hawk (1930)

(From The self-propulsive suit from The Bluff of the Hawk) by Anthony Gilmore:

They had quite a considerable range:

Clad as they were in the latter's self-propulsive space-suits, they were quite capable of reaching Jupiter's Satellite III, only some thirty thousand miles away.