[talanum1 (OP)]

They say a black hole warps spacetime infinitely. Does this mean the distance between space points in a black hole is somehow infinite?

To some alien existing in a black hole it would look like he's in an infinite universe, but we, outside the black hole would think he is in a finite region.

[Halc]

They say a black hole warps spacetime infinitely. Does this mean the distance between space points in a black hole is somehow infinite?

It doesn't mean that, but if you're inside a black hole, there's no spatial limit to your ability to travel. There's no spatial boundary against which you can bump.

To some alien existing in a black hole it would look like he's in an infinite universe

Not true. There's a limit beyond which he cannot see, just like there is for us on Earth. There are places you can see that you can never visit. Things aren't going to suddenly appear larger just because you crossed some arbitrary event horizon.

but we, outside the black hole would think he is in a finite region.

A black hole is not a region of space, finite or otherwise, at least not in the usual coordinate system (where 'meters' are meaningful) to us on the outside.