"policeman," 1859, abbreviation (said to be originally thieves' slang) of earlier copper (n.2), which is attested from 1846, agent noun from cop (v.) "to capture or arrest as a prisoner." Cop-shop "police station" is attested from 1941. The children's game of cops and robbers is attested from 1900.
Each child in Heaven's playground knows each other child by name.
They choose up sides as all take part in some exciting game
Like Hide=and=Seek, Red Rover, Cops and Robbers, Pris'ner's Base.
Our Lord is glad to referee their every game and race.
[John Bernard Kelly, "Heaven is a Circus," 1900]
English has many nouns cop, some archaic or obsolete, many connected more or less obscurely to Old English cop "top, summit," which is related to the source of cup (n.).