Dickinson, 35, made four appearances for the Cheshire outfit after signing in February.
The former Stoke City, Barnsley, Portsmouth and Watford defender made over 400 Football League appearances during his career.
A club spokesperson has said: "Everyone at the club would like to thank Carl for his efforts during his time at the Dabbers and we wish him all the best for the future."
Photo credit: Nantwich Town
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