No banks for bitcoins is available by this time.Just an bitcoin wallet and when comes to deposit rate there is no minimum deposit rate
You are right if someone is looking for a bitcoin bank then for them nowadays there is only one type of banks for bitcoin and that is online wallets in which they can store their bitcoins like banks but they will not get the profit what the fiat banks are giving.
If you look at certain exchanges, then they allow you to store your coins in return for a certain interest rate. I don't see why at some point wallet services can't do the exact same thing. But then again, no service will just hand you over interest without them putting your stored coins to work in one way or another. It requires you to thoroughly check their TOS in order to know what the actual risks are.