A right pain, a lemur, primarily living in tree! (8)
I believe the answer is:
arboreal
'living in tree' is the definition.
(arboreal refers to tree-dwelling animals)
'a right pain a lemur primarily' is the wordplay.
'right' becomes 'r' (common abbreviation).
'pain' becomes 'bore' (both can mean an annoying person).
'primarily' says to take the initial letters.
The initial letter of 'lemur' is 'l'.
'a'+'r'+'bore'+'a'+'l'='ARBOREAL'
(Other definitions for arboreal that I've seen before include "found in branches" , "Resembling a tree" , "Pertaining to trees" , "Tree-dwelling" , "Inhabiting trees" .)