Got three cards I did not want (no rerolls left either) and I wondered: What would happen if I try to ban them all?
Turns out, you can actually ban all of them, and you are not able to continue the game.
The game doesn't crash, but you cannot get out of the cardmenu at all.
Escape does nothing, and you cannot return to the camp in anyway.
Alt+F4 works, and after a restart you are back at where you were before you started the run.
Definitely needs a check so players can't use banish (and freeze?) on the last card.
edit: It's possible to bypass this if you have rerolls left. After banning all cards, rerolling returns two new cards to the selection. Rerolling a second time returns three cards.