1610s, shortened form of opossum. Phrase play possum is first recorded 1822.
To pretend to be dead, a trick used by opossums to defend themselves from predators: “Everyone thought the old con man had died, but it turned out he was just playing possum.” By extension, it also means “to pretend to be asleep, or to lie low”: “Come on, Harry, open your door. You can't play possum forever.”