Backing prison to hold Arab narcissist (9)
I believe the answer is:
egomaniac
'narcissist' is the definition.
(I've seen this in another clue)
'backing prison to hold arab' is the wordplay.
'backing' says the letters should be written in reverse.
'prison' becomes 'cage' (cage is a kind of prison).
'to hold' is an insertion indicator.
'arab' becomes 'Omani' (someone from Oman).
'cage' back-to-front is 'egac'.
'egac' enclosing 'omani' is 'EGOMANIAC'.
(Other definitions for egomaniac that I've seen before include "Obsessively self-absorbed person" , "Abnormally egotistical person" , "number one fan" , "Oneself, one would love" , "Excessively self-concerned person" .)