We show how the language of motivic non-archimedean homotopy theory can be used to define p-adic cohomology theories and prove new results about them. For example, we show how to define solid relative rigid cohomology and deduce a version of Berthelot's conjecture for it (joint work with V. Ertl), and how to construct Hyodo-Kato cohomology, together with its associated Clemens-Schmid chain complex, bypassing log-geometry (joint work with F. Binda and M. Gallauer)