The Chicago Palm Squeezer dates from the 1890's. It was never made in either .32 S&W or .38 S&W; both cartridges are much too long to fit into the gun. AFIAK, those guns were made only in .32, and the round was an extra-short rimfire. The design originated in France. Flayderman estimates production at some 12,800. Cartridges of the World gives the cartridge case length as .398" and the overall length as .645".
There was a similar gun from the same period made by the Minneapolis Firearms Co. It was even smaller than the Chicago gun and was also .32, but centerfire. The round was called the ".32 Protector." COTW gives the case length as .350" and OAL as .555". The even shorter length was necessary since the pistol was smaller. White & Munhall state that the .32 Protector was identical to the 8mm Gaulois.
I can't find any data on velocity, energy, etc., but I can't imagine using either for defense, even if one could find the cartridges, which have probably not been made for a century or so. A .25 Automatic would be a far better choice.
Jim