Could the universe be older and bigger than we can see?

Here's a question that's been puzzling me. If the universe is
expanding, and the further an object (e.g. a galaxy) is away from us,
the faster it is moving away from us. Are there parts of space so far
away from us that it's expanding away from us *faster* than the speed
of light? From what I understand the speed of light limitation doesn't
apply to the expansion of space itself.

Therefore the Cosmic Microwave Background Radiation isn't the cloak
that surrounds the secrets of the Big Bang, but just the curtain around
a part of the universe that is now out of contact with us. An endlessly
expanding universe sure, but one that never had a beginning?

Yousuf Khan