Thank you. The knowledge of members of this forum continues to amaze me.

This has no model number on it other than "LLAMA CAL 380" on the left side directly in front of the take down lever. The proof marks are under that and there are four of them. None are letters, one appears to be a box with something on top of it and an X inside, then a 7, then an open dot, and then a 1.

There is also what I take to be a proof mark on the right side of the slide, just below and behind the rear sight. This matches the last proof mark on the frame and looks like a diamond engagement ring for lack of a better description. Could this be a Stoeger acceptance proof mark?

Update: It's not an engagement ring, it's a bomb with a lit fuse. Which is supposed to have a number inside saying what year it was proofed, but neither do. These are supposed to have a letter P inside indicating that they were proofed, but don't. From doing a bit more research, this looks like a III-A, made in the late 1970s. If I read the proof mark right, it was proofed in 1980. I have to keep editing this as I finally start to get some idea of what I'm seeing. What I thought was a 7 is a poorly stamped Z. Which makes the date code for proofing 1980.

Serial number is located on the right side, 8895xx. It's also marked Stoeger Industries, Hackensack NJ, Made in Spain on the frame above the serial number. Below the serial number is the letter "J".
From the link you included, the "J" date would indicate it was made in 1937. Does that seem possible? It seems unlikely to me as that was during the Spanish Civil War, but I don't know if that has any bearing on the subject.

This seems about the right ball park.