Dynamical Mordell-Lang conjecture for totally inseparable liftings of Frobenius

We prove that if $K$ is a complete algebraically closed non-archimedian valuation field of positive characteristic and $f$ is an endomorphism of ${\mathbb{P}}_K^N$ which is totally inseparable and behaves as the Frobenius on the special fiber, then $f$ satisfies the dynamical Mordell-Lang (DML) property. We also discuss some corollaries and generalizations.