SAN FRANCISCO—The U.S. Court of Appeals for the Ninth Circuit has left in place a Berkeley city ordinance that forces retailers to inform prospective cellphone buyers that carrying their devices in certain ways can cause exposure to radio-frequency radiation that exceeds federal limits.

CTIA, a wireless industry trade association, had brought on Theodore Olson of Gibson, Dunn & Crutcher to bring a First Amendment challenge to the local law and to argue that federal regulations pre-empt it.