The only way I know of getting rid of carpenter ants is to destroy there nest. I would also power wash your palm with a hose and try to clean your palm of the ants.
And in the meantime keep your palm happy. Give it some palm fertilizer and water and let it grow as fast as possible so it recovers.
Good luck!

Sounds like cold damage and the ants just moved into the dead part to live. You can spray with an insecticide and that will kill them. All the other non poison ways don't work very well in my opinion.

I recommend permethrin at one teaspoon to one gallon water. Soak the nest and the roots of the tree. It is harmless to palms but deadly to ants.