A planar set $G \subset {\bb R}^2$ is constructed that is bilipschitz equivalent to ( $G, d^z$ ), where ( $G, d$ ) is not bilipschitz embeddable to any uniformly convex Banach space. Here, $z \in (0, 1)$ and $d^z$ denotes the $z$ th power of the metric $d$ . This proves the existence of a strong $A_{\infty}$ weight in ${\bb R}^2$ , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of ${\bb R}^2$ .