The third doctor's first car was called Bessie, later in his life though he got another car nicknamed the who mobile
the Doctor's car during his exile on Earth was Bessie. Used mainly by the Third Doctor, the Fourth and Seventh also used it, as well as other Doctors off screen. He also had another car (that hovered and could also fly). It was unnamed on screen, but referred to by Jon Pertwee as the "Whomobile".
The "Whomobile" was Jon Pertwee's own property, so could not be used by later incarnations of the Doctor.