There are already coins that offer crypto currency banking services.
Minexcoin you can park your coins into the MinexBank without having to give over your private keys.
Another one is bankex they are in the middle of their ICO at the moment, I decide to invest 2 eth into their awesome project.
The OP was unequivocal and ,succinct about his question and it was bitcoin bank that he was asking for not some altcoin that we don't know off. There's currently no bitcoin bank anywhere that i know of but maybe in the future when bitcoin education has spread abroad we may have them in our communities.
Yes, that is really true. This time there is no bitcoin bank to be seen everywhere in the world but sooner or later, I believe that bitcoin bank will exist depending on the needs of the people who are using bitcoin as a means of payment in their respective countries.