Loudly auction off part of prison (4)
I believe the answer is:
cell
'part of prison' is the definition.
('cell' is part of 'prison')
'loudly auction off' is the wordplay.
'loudly' indicates a 'sounds like' (homophone) clue.
'auction off' becomes 'sell'.
'sell' is a homophone of 'CELL'.
(Other definitions for cell that I've seen before include "Monk's small room" , "Place of incarceration in body unit" , "Dungeon" , "Chamber" , "Spreadsheet unit" .)