An unbounded convex set with the fixed point property for nonexpansive mappings

We construct an equivalent norm on $\mathbb{R}\oplus {\mathrm{\ell }}_2$ and a nonempty closed convex unbounded subset having the fixed point property for nonexpansive self-mappings. The ambient renormed space has the fixed point property on bounded closed convex subsets. The main point is a geometric localization estimate at a distinguished point of the unbounded set; the fixed point conclusion then follows from the Kirk–Ray localization theorem.