  The purpose of this paper is to prove a basic $p$-adic comparison theorem for
smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a
$p$-adic field: $p$-adic pro-\'etale cohomology, in a stable range, can be
expressed as a filtered Frobenius eigenspace of de Rham cohomology (over
$\bf{B}^+_{\rm dR}$). The key computation is the passage from absolute
crystalline cohomology to Hyodo-Kato cohomology and the construction of the
related Hyodo-Kato isomorphism. We also "geometrize" our comparison theorem by
turning $p$-adic pro-\'etale and syntomic cohomologies into sheaves on the
category ${\rm Perf}_C$ of perfectoid spaces over $C$ (this geometrization will
be crucial in our proof of the $C_{\rm st}$-conjecture in the sequel to this
paper).
