Arab in English golf and athletic club is conceited individual (9)
I believe the answer is:
egomaniac
'conceited individual' is the definition.
(an egomaniac is conceited or arrogant)
'arab in english golf and athletic club' is the wordplay.
'arab' becomes 'Omani' (someone from Oman).
'in' means one lot of letters goes inside another.
'english golf' becomes 'eg' (this might be a standard abbreviation of which I'm unaware).
'and' says to put letters next to each other.
'athletic club' becomes 'AC' (abbreviation).
'eg'+'ac'='egac'
'omani' inserted within 'egac' is 'EGOMANIAC'.
'is' is the link.
(Other definitions for egomaniac that I've seen before include "Abnormally egotistical person" , "Oneself, one would love" , "number one fan" , "Excessively self-absorbed person" , "who's full of himself" .)