Branch that is not on fire (8)
I believe the answer is:
offshoot
'branch' is the definition.
('offshoot' can be a synonym of 'branch')
'not on fire' is the wordplay.
'not on' becomes 'off' ('off' is the opposite to 'on').
'fire' becomes 'shoot' (shooting is a kind of firing).
'off'+'shoot'='OFFSHOOT'
'that is' is the link.
(Other definitions for offshoot that I've seen before include "Small branch growing from the main stem" , "Branch attached to the main stem" , "sequel?" , "Branch, derivative" , "Subsidiary product or branch" .)