Green’s Dictionary of Slang

(orig. milit.) a cigarette end, esp. one that is large enough to be relit.

1937		R. Westerby Wide Boys Never Work (1938) 214: Look at him: with half-smoked ‘dimp’ hanging from his upper lip.
1958		(con. 1940s) B. Behan Borstal Boy 107: A couple of times [...] I got in a little dimp for Browny.
1974		(con. 1922) A. Pierrepoint Executioner 48: I was standing at the corner of the street with the rest of the lads, passing round the dimps, or cigarette ends.
[2009		T. Black Gutted 85: I threw my dowp out of the window].
2024		J. Meades Empty Wigs (t/s) 478: [A] man in rags sitting on the doorstep of an Ugford-on-Roden shop smoking a dimp.