Rigid cohomology via the tilting equivalence

We define a de Rham cohomology theory for analytic varieties over a valued field $K^\flat$ of equal characteristic $p$ with coefficients in a chosen untilt of the perfection of $K^\flat$ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic.