One given lead in Mexico? (9)

I believe the answer is:

chihuahua

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'one given lead' is the definition.
(breed of dog)

I don't understand the remainder of the clue.

(Other definitions for chihuahua that I've seen before include "Dog breed from Mexico" , "Breed of tiny dog" , "Breed of small, short-haired dog" , "that's running very low?" , "Mexican state -- breed of dog" .)

I've seen this clue in The Telegraph.

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