Let $A$ be an abelian variety over a complete non-Archimedean field $K$. The universal cover of the Berkovich space attached to $A$ reflects the reduction behaviour of $A$. In this paper the universal cover of the universal vector extension $E(A)$ of $A$ is described. In a forthcoming paper ( arXiv:2007.04659), this will be one of the crucial tools to show that rigid analytic functions on $E(A)$ are all constant.