Are you sure it was FAG 32? Should be FAG 39. "FAG" on Carcanos is the abbreviation for Fabbrica d'Army Gardone and a date..The "FAG" marking is often seen on rebuilt rifles.
It probably stands for the same on a Beretta but I have no proof of this. It is odd that a Beretta would have a Gardone factory marking. The "FAG" has only been noted on Model 1934 (380) Beretta pistols with slide dates of 1939 and 1940, all are also "RE" marked.
Which bring up another question, why would there be a FAG 39 date on a 1940 pistol? Could be they just kept using the FAG 39 stamp in 1940. But why was the Gardone Weapons Factory marking Beretta pistols in 1939 and 1940. I don't see any reason for Gardone to be rebuilding Berettas, unless they were helping them with production in 1939/40. There are a lot of other oddly marked Model 1934 pistols in 1939 and 1940 in the "FAG" pistol serial range, all the pistols in 1939/40 are not "FAG" marked.
If you see that pistol again double check the FAG 32 marking.