Will we ever be able to see the beginning of the universe?

As I understand it, when I look at the sun, I don’t see the sun as it is now, rather I see the light of it from 8 minutes ago, and when I look at our closest star after the sun, I am seeing it as it was 4 years ago.

At the moment, we can see about 13 billion years back in time when we look deep in to the universe. Seeing distant galaxies not as they are, but as they used to be.

From what I understand, we can’t see further because the light has not reached us yet. However, in another billion years time, we will be able to see a billion light years further than we can now.

The question is, will there ever come a time, when there is light reaching us from so far away, that it allows us to see what the universe was like at the time of the big bang or just after it?

I don’t mean background radiation or anything like that, I mean seeing as in traditionally seeing the visible light with your eyes.