Let $X$ be an $n$-dimensional smooth projective variety of geometric origin from a $d$-dimensional family over a perfect field $k$ of characteristic $p>0$. We assume that resolution of singularities and embedded resolution of singularities hold over $k$ in dimension $d+n$. Let $L$ be an ample line bundle on $X$ and let $\mu$ be a positive Radon measure supported on the skeleton of a projective SNC-model of $X$ with $\mu({X^{{\mathrm{an}}}})=\deg_{L}(X)$. Then the non-archimedean Monge–Ampère equation (1.1) has a continuous semipositive metric ${\|\ \|}$ on $L^{{\mathrm{an}}}$ as a solution.