Grey has overcome rise of red squirrel (7)
I believe the answer is:
hoarder
'squirrel' is the definition.
(squirrels hoard nuts)
'grey has overcome rise of red' is the wordplay.
'grey' becomes 'hoar' (synonyms).
'has overcome' means one lot of letters go next to another (some letters go over or above others).
'rise of' says the letters should be written backwards (in a down clue, letters go up).
'red' in reverse letter order is 'der'.
'hoar'+'der'='HOARDER'
(Other definitions for hoarder that I've seen before include "Saver" , "One secretly stores money or valuables" , "Accumulator" , "Miser; magpie" , "One who accumulates things" .)