A framework $(G,p)$ in Euclidean space $\mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^{2} \setminus \{0\}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $e\in E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.