If space isn't part of the Universe, what is the Universe expanding into?
There is nothing required for the universe to expand into, because all space and time - any reality that we know of - is coming into existence as the universe expands.
It is a common misconception to think that there is something larger than the universe, and that the universe is expanding into it (perhaps to fill it at some point?).
Current belief is that time and space came into existence at the moment of the Big Bang, and that as a result, there was no such thing as "before the Big Bang". In much the same way, there is no such thing as "beyond the universe".
One of my favourite Stephen Hawking quotes is this one:
"Although the universe doesn't have an end, it had a beginning in the Big Bang. One might ask what is before that but the answer is that there is nowhere before the Big Bang just as there is nowhere south of the South Pole."
Likewise, there is nowhere outside the universe.