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For an abelian variety $A$ over an algebraically closed non-archimedean field$K$ of residue characteristic $p$, we show that the isomorphism class of thepro-\'etale perfectoid cover $\widetilde A=\varprojlim_{[p]}A$ is locallyconstant as $A$ varies $p$-adically in the moduli space. This gives rise to apro-\'etale uniformisation of abelian varieties as diamonds\[A^\diamond=\widetilde A/T_pA\] that works uniformly for all $A$ without anyassumptions on the reduction of $A$.  More generally, we determine all morphisms between pro-finite-\'etale coversof abeloid varieties. For example, over $\mathbb C_p$, all abeloids can beuniformised in terms of universal covers that only depend on the isogeny classof the semi-stable reduction over $\bar{\mathbb F}_p$.

Pro-\'etale uniformisation of abelian varieties