Constituent of Britain, say, lives life on the edge (4)
I believe the answer is:
isle
'constituent of britain' is the definition.
(one of the British Isles)
'lives life on the edge' is the wordplay.
'lives' becomes 'is' ('be' can be a synonym of 'live').
'on the edge' means to remove the middle letters (only the letters at the edges).
'life' with its middle removed is 'le'.
'is'+'le'='ISLE'
'say' is the link.
(Other definitions for isle that I've seen before include "being impossible without water" , "Man's one" , "what Donne implied he wasn't" , "Small area of land in water" , "description of Britain" .)