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I don't see that option in Word.

In Excel I used &date for the footer but that doesn't seem to work in Word.
 
Pity. I use it all the time in the desktop version of Word.

Even editing it online (from onedrive.live.com) doesn't give you the ability to insert a date field. The date field I have added from the desktop client remains so it sees it, you just cannot edit it.
 
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