They have their own Isle, pets (4)

I believe the answer is:

dogs

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'pets' is the definition.
(I've seen this before)

I don't understand the rest of the clue.

(Other definitions for dogs that I've seen before include "Shadows" , "eg 27 terriers" , "Canines" , "Hounds" , "Pursues" .)

I've seen this clue in the Evening Standard.

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