In paper [B. Dragovich, A $p$-adic matter in a closed universe, Symmtry 2022, 14,73 ], we introduced $p$-adic scalar particles, which have their origin in $p$-adic strings, i.e. strings with $p$- adic world sheet. These particles have nonlinear and nonlocal dynamics, without tachyons and ghosts. We considered cosmological solutions and shown that mass of $p$-adic particles is related to the cosmological constant. Here we discuss possible role of $p$-adic scalar particles as dark energy and dark matter.