  We develop a theory of \'etale parallel transport for vector bundles with
numerically flat reduction on a $p$-adic variety. This construction is
compatible with natural operations on vector bundles, Galois equivariant and
functorial with respect to morphisms of varieties. In particular, it provides a
continuous $p$-adic representation of the \'etale fundamental group for every
vector bundle with numerically flat reduction. The results in the present paper
generalize previous work by the authors on curves. They can be seen as a
$p$-adic analog of higher-dimensional generalizations of the classical
Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide
new insights into Faltings' $p$-adic Simpson correspondence between small Higgs
bundles and small generalized representations by establishing a class of vector
bundles with vanishing Higgs field giving rise to actual (not only generalized)
representations.
