Fish that's fat dropping by (4)
I believe the answer is:
chub
'fish' is the definition.
(chub is a kind of fish)
'fat dropping by' is the wordplay.
'fat' becomes 'chubby' (chubby means fat or plump).
'dropping' is a deletion indicator.
'chubby' with 'by' taken out is 'CHUB'.
'that's' is the link.
(Other definitions for chub that I've seen before include "River fish; rhymes with public house" , "Thick-bodied European fish" , "European freshwater fish" , "Plump fish" , "Thick-bodied river fish" .)