Of course they can. A bank would have no problems giving out loans given the pay is coming eventually.
These banks do not exist.
I'm not a Federal employee and I don't know the details, but there's a banner on my bank's (Chase) website (after you log in) suggesting that they have mechanisms to assist those who aren't getting paid.
What those are, I have no idea. It may well be high interest loans or no interest loans. Or it may waive fees on overdrafts. Again, I don't know. But banks are taking note and communicating with their customers about it.