This is a Sauer 38H, in .32 ACP:
It's in an apparently military-issue flap holster, complete with a second magazine:
The magazines are serial-numbered to the gun.
The gun has a concealed-hammer DA/SA type action. The lever above the magazine release is a decocker, which functions just like a modern Sig, at least on decocking. But unlike modern guns, the 38H's decocker also functions as a cocker: Pushing it down on an uncocked gun will cock the hammer. (It's probably best to not use it, as obtaining replacement springs and parts is problematical.) Like a PPK, there is no slide release. Unlike a PPK or modern guns, there is no slide-locking mechanism.
The original guns didn't have a safety, but the Luftwaffe wanted one, so like any other good government contractor, Sauer added one. It could be because the Luftwaffe issued them to paratroopers and the safety was added protection from an accidental discharge in a hard landing. Apparently, German paratroopers jumped with their long guns in separate equipment canisters; the 38H (and other handguns) gave them a weapon to jump with.
The lever at the top of the trigger guard is for disassembly. Pull down on it; it functions like pulling down on the trigger guard of a PPK:
After you do that, it field-strips in the exact same way as a PPK. The bore is bright and sharp.
The internal markings of the slide match the serial number. On the front web of the trigger guard, there is an "eagle c" acceptance stamp.
The safety and the acceptance marking would hint at the gun being of post-1940 manufacture, but not terribly much later.
The "c" marking indicates that the gun was destined for police use. The holster apparently is of Luftwaffe-type, as it seems that the Reich cops used black leather holsters.
There are no import markings of any kind. Interarms brought in a bunch after the war and pre-68; those apparently are marked "7.62mm" on the right side of the slide. This one isn't so marked. It may have been a "GI-bring back", but there is no way I know of to tell that.
I hope to be able to take it to the range, after applying a little bit of lubrication.
UPDATE:
Range report.