Stiff with cold, short man in a river (7)
I believe the answer is:
cadaver
'stiff' is the definition.
('cadaver' can be a synonym of 'stiff')
'cold short man in a river' is the wordplay.
'short' indicates taking the first letters.
'man' becomes 'dave' (I've seen this before**).
'in' means one lot of letters goes inside another.
'river' becomes 'r'.
The initial letter of 'cold' is 'c'.
'a'+'r'='ar'
'dave' inserted inside 'ar' is 'adaver'.
'c'+'adaver'='CADAVER'
'with' is the link.
(Other definitions for cadaver that I've seen before include "Dead human body" , "Stiff" , "Dead body, corpse" , "Human corpse" , "Human body dissected by medical students" .)