  Our goal is to study $p$-adic local systems on a rigid-analytic variety with
semistable formal model. We prove that such a local system is semistable if and
only if so are its restrictions to the points corresponding to the irreducible
components of the special fiber. For this, the main body of the paper concerns
analytic prismatic $F$-crystals on the absolute logarithmic prismatic site of a
semistable $p$-adic log formal scheme. Analyzing Breuil-Kisin log prisms, we
obtain a prismatic purity theorem and deduce the above purity theorem for
semistable local systems.
