Banks are NOT Federal facilities they are private businesses and are not on the Florida or Federal restriction list so you can carry in banks in Florida.
I am not certain what the comments about signs are about??? Florida has no "no gun" sinage laws that I am aware of, so even if a business (including a bank) has a sign stating no guns, you are not violating any law if you carry into that business. The business owner could ask you to leave their premises, and if you refuse you could be arrested for trespass, but that is about the extent of it.
Signs don't mean anything in Florida, which is why no one posts them.
I will note for the record however that I like the sign at the entrance of our new Gander Mountain store which reads "NO LOADED FIREARMS ALLOWED" ... then notes this does not apply to holders of a valid concealed weapons permit.