Bitcoin banks do exist but they are a monumentally stupid thing to put your money in.  The reason for this is that because Bitcoin is not very established nor is the price stable, there is no bank that you can genuinely trust to continue paying out forever, just like an exchange.  The chance of any given "Bitcoin bank" collapsing partway through your investment with your money still in it is dramatically higher than the interest that you will earn from it as a Bitcoin bank would act like an exchange, just giving the promise that you will get your Bitcoin back (you wouldn't own the Bitcoin anymore).