Name of female oil-producer ... (5)
I believe the answer is:
olive
'name' is the definition.
'of female oil-producer' is the wordplay.
'of' becomes 'o''.
'female oil producer' becomes 'live' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'o'+'live'='OLIVE'
Can you help me to learn more?
(Other definitions for olive that I've seen before include "girl" , "Mediterranean tree-fruit, nice in cocktail" , "Greyish-green; fruit" , "woman" , "Bitter fruit" .)