Caught that fellow longing to climb cleft in rock-face (7)
I believe the answer is:
'cleft in rock-face' is the definition.
(term for a type of rock formation)
'caught that fellow longing to climb' is the wordplay.
'caught' becomes 'c' (cricket abbreviation).
'that fellow' becomes 'him' ('him' refers to a man).
'longing' becomes 'yen' (yen can mean a desire or longing for something).
'to climb' is a reversal indicator (I've seen 'climbing' mean this (in a down clue, letters climb up)).
'yen' reversed gives 'ney'.
(Other definitions for chimney that I've seen before include "Smoke pipe; rock cleft" , "Smoker on the roof?" , "Smoke outlet" , "Father Christmas's entry?" , "Flue" .)