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