I'm not certain, but I would almost bet that a dome light in a 1939 would have been an aftermarket addition. If it was from the factory, I would think that it would not have a door switch, but rather turned on by a switch on the light itself.
Well, it so happens you're absolutely correct! the light must be an aftermarket addition since it has a switch on the light itself, rather than a door switch... I overlooked that detail... Light is marked Arrow 36...