I have been carrying into banks (in Florida, where as already stated, it is legal) for years. Conceal properly and there should be no problem.
I even went to the bank to have my CCL renewal notarized the last time around. The guy who helped me said he had been doing a lot of them recently, and seemed quite comfortable with it. He didn't ask if I was carrying at the moment, but I got the impression that he assumed I was, and he would have been correct.