According to www.gunsworld.com, it was a Browning 1910, Belguim made, .32 ACP
Live pic.
There appears to be some uncertainty on whether it was a .32 or a .380, both were being manufactured at the time.
Incredibly enough, I have one just like it. My late father used it in his espionage work (AFOSI) in W. Germany and North Africa in the early 1950s. He always said he liked it because it fit neatly under a suitjacket without betraying a tell-tale lump. It got held to more than one Polish Intelligence and Stasi agent's forehead . . .
BTW, it still shoots just fine. But I wouldn't trust its accuracy more than about 30 paces. Then again, I'm the worst pistol shot in the world.