Asian lady's heading off before one (5)
I believe the answer is:
omani
'asian' is the definition.
(Oman is an Asian country)
'lady's heading off before one' is the wordplay.
'lady's' becomes 'woman'.
'heading off' means to remove the first letter.
'before' says to put letters next to each other.
'one' becomes 'i' (Roman numeral).
'woman' with its first letter taken off is 'oman'.
'oman'+'i'='OMANI'
(Other definitions for omani that I've seen before include "Someone from Muscat" , "Gulf Arab" , "Citizen of a country, capital Muscat" , "Gulf native" , "One from Muscat" .)