Dissect a cat in an island quarter (9)

I believe the answer is:

anatomise

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'dissect' is the definition.
(anatomising is a kind of dissecting)

'a cat in an island quarter' is the wordplay.
'a' becomes 'an'.
'cat in an island quarter' becomes 'atomise' (I can't explain this - if you can you should give a lot more credence to this answer).
'an'+'atomise'='ANATOMISE'

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(Another definition for anatomise that I've seen is " Closely examine".)

I've seen this clue in The Times.

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