We construct a functor from the category of $p$-adic local systems on a smooth rigid analytic variety $X$ over a $p$-adic field to the category of vector bundles with a connection on $X$, 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 results about the $p$-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. Joint work with Xinwen Zhu.