High-flyer's hunting dog slipped off its lead (5)

I believe the answer is:

eagle

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'high-flyer's' is the definition.
(eagles fly high)

I cannot understand how the rest of the clue works.

(Other definitions for eagle that I've seen before include "high-flier" , "Erne, perhaps" , "Valuable piece" , "It normally takes two or three shots to shoot this" , "Keen-eyed bird" .)

I've seen this clue in The Independent.

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