British businessman in a pickle, short of time (7)
I believe the answer is:
branson
'british businessman' is the definition.
(Richard Branson, founder of Virgin)
'a pickle short of time' is the wordplay.
'a pickle' becomes 'Branston' (Branston is a pickle brand).
'short of' suggests deleting specific letters.
'time' becomes 't' (abbreviation).
'branston' with 't' taken away is 'BRANSON'.
'in' is the link.
(Other definitions for branson that I've seen before include "Richard -, Virgin head" , "Sir Richard --, British entrepreneur" .)