Green’s Dictionary of Slang

nap n.4

[nab n.1 (3)]

a hat.

[ ‘Hat of Other Days’ in Lloyd’s Pickwickian Songster 7: The Hat of other days is faded, [...] By many a sleepy spot ’tis shaded, / It wants a nap at last].
[UK]Hotten Dict. of Modern Sl. etc.
[UK]Sl. Dict.