iven an étale Zp-local system of rank n on an algebraic variety X, continuous cohomology classes of the group GLn(Zp) give rise to classes in (absolute) étale cohomology of the variety with coefficients in Qp. These characteristic classes can be thought of as p-adic analogs of Chern-Simons characteristic classes of vector bundles with a flat connection. If X is a smooth variety over a p-adic field, and the local system is de Rham, then its characteristic classes turn out to be related to Chern classes of the graded quuotients of the Hodge filtration on the associated vector bundle with connection. This relation can be established through considering an analog of Chern classes for locally free sheaves of OX-modules on the pro-étale site of X. This is a joint work with Lue Pan.