  We study the topology of the moduli space of tropical unramified
$\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$ where $p$ is a
prime number. We use recent techniques by Chan--Galatius--Payne to identify a
contractible subcomplex of the moduli space. We then use this contractibility
result to show that this moduli space is simply connected for all $g$ and $p$.
In the case of genus $2$ we fully determine the homotopy type of this moduli
space for all primes $p$. This work is motivated by prospective applications to
the top-weight cohomology of the space of prime cyclic \'etale covers of smooth
algebraic curves.
