  We construct a functor from the category of p-adic etale local systems on a
smooth rigid analytic variety X over a p-adic field to the category of vector
bundles with an integrable connection over its "base change to B_dR", which can
be regarded as a first step towards the sought-after p-adic Riemann-Hilbert
correspondence. As a consequence, we obtain the following rigidity theorem for
p-adic local systems on a connected rigid analytic variety: if the stalk of
such a local system at one point, regarded as a p-adic Galois representation,
is de Rham in the sense of Fontaine, then the stalk at every point is de Rham.
Along the way, we also establish some basic properties of the p-adic Simpson
correspondence. Finally, we give an application of our results to Shimura
varieties.
