I.e. place being rebuilt around me bit by bit (9)
I believe the answer is:
piecemeal
'bit by bit' is the definition.
('piecemeal' can be a synonym of 'bit by bit')
'i e place being rebuilt around me' is the wordplay.
'being rebuilt' indicates an anagram.
'around' is an insertion indicator.
'i'+'e'+'place'='ieplace'
'ieplace' anagrammed gives 'pieceal'.
'pieceal' going around 'me' is 'PIECEMEAL'.
(Other definitions for piecemeal that I've seen before include "taken bit by bit" , "Bit by bit; unsystematic" , "Unsystematically" , "a little at a time" , "A bit at a time" .)