Religious teacher grew dizzy on island (5)
I believe the answer is:
swami
'religious teacher' is the definition.
(I've seen this before)
'on island' is the wordplay.
'on' becomes 'swam' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'island' becomes 'i'.
'swam'+'i'='SWAMI'
'grew dizzy' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for swami that I've seen before include "Religious master" , "Hindu leader" , "Hindu mystic" , "Guru" , "Hindu holy man" .)