  We show how to functorially attach continuous $p$-adic representations of the
profinite fundamental group to vector bundles with numerically flat reduction
on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes
results by Deninger and Werner for vector bundles on smooth algebraic
varieties. Our approach uses fundamental results on the pro-\'etale site of a
rigid analytic variety introduced by Scholze. This enables us to get rid of the
smoothness condition and to work in the analytic category. Moreover, under some
mild conditions, the functor we construct gives a full embedding of the
category of vector bundles with numerically flat reduction into the category of
continuous $\mathbb{C}_p$-representations. This provides new insights into the
$p$-adic Simpson correspondence in the case of a vanishing Higgs-field.
