Squeeze out food store with extremely fast mail (7,8)

I believe the answer is:

express delivery

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'fast mail' is the definition.
The definition suggests a singular noun which matches the answer.

'squeeze out food store with extremely' is the wordplay.
'squeeze out' becomes 'express' (I am not sure about this - if you are sure you should believe this answer much more).
'food store' becomes 'deli' (I've seen this in other clues).
'with' means one lot of letters go next to another.
'extremely' becomes 'very' (very can mean extremely or considerably).
'express'+'deli'+'very'='EXPRESS DELIVERY'

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I've seen this clue in the Sydney Morning Herald.

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