Exist, drinking water, finally alive (9)

I believe the answer is:

breathing

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

I cannot understand how the rest of the clue works.

(Other definitions for breathing that I've seen before include "could be rough for Hellenes" , "It's inspiring" , "showing a sign of life" , "Respiring" , "(Of wine) exposed to fresh air" .)

I've seen this clue in The Sunday Times.

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