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.