I have seen a type of banking proposal where the bank always borrows from the central bank at x% and issues loans that way and just passes the x% cost to anyone holding the money. That means no deposits were required to begin with. If anything, cash means that the person who withdrew money would be stuck with the borrowing fee until they deposit their cash back in.

So yeah although this is still theoretical, it is indeed possible to do banking without having a single customer cash deposit.