River rodent, the first to go extinct (4)
I believe the answer is:
ouse
'river' is the definition.
(Yorkshire river)
'rodent the first to go extinct' is the wordplay.
'rodent' becomes 'mouse' (mouse is a kind of rodent).
'the first to go extinct' means to remove the first letter.
'mouse' with its first letter taken away is 'OUSE'.
(Other definitions for ouse that I've seen before include "River flowing through York" , "River that joins the Trent to form the Humber" , "bird has lost its tail" , "N English river (and S English river)" , "Name of river in Sussex or Yorkshire" .)