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We show that there exists a non-weakly compact, closed, bounded, convex subset W of the Banach space of convergent sequences ( c , ‖ ⋅ ‖ ∞ ) , such that every nonexpansive mapping T : W ⟶ W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space X isomorphic to c 0 , for which W has the fixed point property for nonexpansive mappings. We also prove that the sets W may be perturbed to a large family of non-weakly compact, closed, bounded, convex subsets W q of ( c , ‖ ⋅ ‖ ∞ ) with the fixed point property for nonexpansive mappings; and we discuss similarities and differences with work of Goebel and Kuczumow concerning analogous subsets of ℓ 1 . [ABSTRACT FROM AUTHOR]