It's a bit more complicated than that. General relativity allows you to pick any reference frame, even one that is bent, stretched or distorted in some other way, and do your calculations in that reference frame.
You could pick a "normal" reference frame that obeys the special theory of relativity: speed of light constant everywhere, nothing can go faster, etcetera. Nothing wrong with that, but this turns out to be impractical: we have to pick some place to consider as the center of the universe (for example some place in our immediate neighborhood), and then find that the rest of the universe is moving away at very high speeds, approaching (but not exceeding) the speed of light. This means those galaxies are shrunk in the direction of their motion (Lorentz contraction) and time passes more slowly for them (time dilation). The further you "look" (the infinitely quick kind of looking which you can only do inside a theoretical model, not having to wait for light to get here so we can actually see stuff), the more things are shrunk and the slower time is ticking. At a distance of the speed of light ("c") times the age of the universe, things approach the speed of light and time is passing so slowly that the Big Bang is only just happening right now. In this way of describing the universe, with these coordinates the universe actually fits in a finite sphere around us.
That's a perfectly valid set of coordinates, but I think you'll agree it's not very practical. So physicists invented the cosmological model: imagine a bunch of clocks everywhere in the universe, flying at the same speed as the expansion of the universe (i.e. the same speed as average galaxies in that neighborhood) and ticking at whatever rate the local clocks are ticking at (not synchronized to ours). We define time at any place in the universe as being whatever is indicated by those clocks, not ours. So in effect we change the very definition of simultaneity, moving things from the future into today simply by changing the labeling. Also, imagine measuring sticks available everywhere in the universe, but just like the clocks flying at the same speed as the local expanding universe. To measure distances, we use those sticks instead of our own.
If we now measure everything using local (Lorentz-contracted) sticks and local (time-dilated) clocks, the universe looks completely different. It is truly infinite, the same age everywhere, and distant objects are no longer flat Lorentz-contracted pancakes but look the same as objects in our neighbourhood. Note that this is not a different universe, it's the same one but with different labels stuck onto objects.
Now, with this set of coordinates, it turns out that rays of light don't travel at a fixed speed "c" relative to us, but relative to the local clocks and sticks we used to define the coordinate system. It is still true that nothing can go faster than (local) light, i.e. you cannot overtake a ray of light, but a distant object certainly can move away from us faster than the speed of a ray of light in our neighbourhood. And if some alien over there were to try and shoot a laser beam our way, that light would never reach us because it is traveling towards us at the speed of light relative to the local "space" which is moving away from us faster, like a cosmic conveyor belt. Note that this conveyor belt is not real, it's just a product of our mathematical trickery refefining distances and times.
Of course you might wonder what happens to that alien laser beam in the first coordinate system, where rays of light all travel at the same speed relative to us. Well, in that system, the aliens don't exist yet because time in that part of space is moving very slowly (and has been moving slowly ever since the big bang). And since that part of space is still accelerating away from us ever faster and closer to the speed of light, local time comes to an asymptotic halt before the aliens ever get a chance to shoot that laser.
"Space itself" is just whatever we define it to be. By changing coordinates, we can move things from the past into the future or even into "never". It doesn't matter, it's just math(s), the end result is that we will never see that laser and we will never be able to reach that galaxy either.