  In this article we use techniques from tropical and logarithmic geometry to
construct a non-Archimedean analogue of Teichm\"uller space
$\overline{\mathcal{T}}_g$ whose points are pairs consisting of a stable
projective curve over a non-Archimedean field and a Teichm\"uller marking of
the topological fundamental group of its Berkovich analytification. This
construction is closely related to and inspired by the classical construction
of a non-Archimedean Schottky space for Mumford curves by Gerritzen and
Herrlich. We argue that the skeleton of non-Archimedean Teichm\"uller space is
precisely the tropical Teichm\"uller space introduced by Chan-Melo-Viviani as a
simplicial completion of Culler-Vogtmann Outer space. As a consequence, Outer
space turns out to be a strong deformation retract of the locus of smooth
Mumford curves in $\overline{\mathcal{T}}_g$.
