Banks are NOT considered US govt or federal property!
Many citizens see the FDIC signs & think it means it's federal property. It just means that business or bank is insured. You can carry a loaded concealed firearm there if you have a valid license/permit. Check with your local city/state laws too.
I had a co-worker in NW Florida drone on & on about carrying weapons into the BoA branch while doing a storm recovery detail in 2004. Just be cool & mature and no one will say anything.
Now a USPS location is different, they do have no firearms laws. If you are packing heat there, you could face a federal arrest & conviction.