Well Lying to the FBI and other federal agents is direct violation of the law

Lying to general police while not direct violation of the law, can be (and has been) prosecuted under several laws such as obstruction of justice, also several states have statues making false or misleading statements to police or public officials during in official investigation. Then there are also laws that prohibiting lying in specific context like your name, address, and other details.

You can say nothing, but you cannot make a false statement