Green’s Dictionary of Slang

possum v.

[SE possum + possum n. (1), i.e. the habits of the animal]

[early 19C+] to dissemble; to feign sickness; thus possum trick n., a feigning of injury to lure an intended robbery victim.

In phrases

possum-foot (v.)

[1960s] to tread quietly.

possum up (v.)

[1950s] to be quiet.