  Let $\Omega$ be the complement of a connected, essential hyperplane
arrangement. We prove that every dominant endomorphism of $\Omega$ extends to
an endomorphism of the tropical compactification $X$ of $\Omega$ associated to
the Bergman fan structure on the tropicalization of $\Omega$. This generalizes
a previous result by R\'emy, Thuillier and the second author which states that
every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$
extends to an automorphism of the successive blow-up of projective space at all
$\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact
the minimal wonderful compactification by de Concini and Procesi, which
coincides with $X$ by results of Feichtner and Sturmfels. Whereas the previous
proof is based on Berkovich analytic geometry over the trivially valued finite
ground field, the generalization discussed in the present paper relies on
matroids and tropical geometry.
