You have a 38 Special M&P that was actually made just post WWII if I am reading the book correctly.
The serial number includes the prefix letter "C" so it is C87785.
My 3rd edition of the Catalog of S&W says the C prefix serial numbers started with C1 in 1948 and ended with C999999 in 1967 so I'm guessing that yours would probably have been made in 1948 or 1949. The finish and the stocks both look to be in excellent shape.
Didn’t even see the “C” until you pointed it out. Just had it cleaned and test fired last week (it was pretty dust covered despite being kept in an old Kolpin gun case). It cleaned up really nice though. Really appreciate the info. Thank you.