Crossword Genius

Girl’s granny, extremely canny (5)

Ross

I believe the answer is:

nancy

'girl's' is the definition.

'granny extremely canny' is the wordplay.
'granny' becomes 'nan'.
'extremely' suggests removing the centre (extreme, outside letters only).
'canny' with its centre removed is 'cy'.
'nan'+'cy'='NANCY'

(Other definitions for nancy that I've seen before include "Birthplace of Edmond de Goncourt" , "City in Lorraine" , "Bill Sikes's girl (Dickens)" , "Lorraine's part" , "Dickensian character who died" .)

I've seen this clue in The Times.
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