I think there is no need, this is because if the existence of the bitcoin banks then will make bitcoin same with fiat money. In addition, the price will be in full control of the bank by making it unprofitable.
It is possible, but on the second thought, they can't dictate the price of bitcoin as many will protest when that happens. Anyway, I think we need a bitcoin bank, but they should have a purpose same as the real life banks. They should offer loans and other things that real banks offered. If they would just function like exchanges,
then those banks will be a failure 
They might fail as banks, but not necessarily as exchanges
In fact, there are a few projects which have the word bank as part of their name (e.g. ChronoBank and CoinsBank). I don't know anything about the former (ChronoBank) but the latter (CoinsBank) started as an exchange (back then it was called Bit-X). As said many times already, loaning bitcoins is not a viable idea presently, but the major reason why they began pretending to be a bank was issuing Bitcoin payment cards. Some established web wallets (like Coinbase and Xapo) do the same, but they don't pretend to be banks. Perhaps, there are other services offered by web wallets and not by exchanges which could be construed as bank services that would allow us to consider them as Bitcoin banks, to a certain degree