  We construct a relative Hodge-Tate spectral sequence for any smooth proper
morphism of rigid analytic spaces over a perfectoid field extension of $\mathbb
Q_p$. To this end, we generalise Scholze's strategy in the absolute case by
using smoothoid adic spaces. As our main additional ingredient, we prove a
perfectoid version of Grothendieck's "cohomology and base-change". We also use
this to prove local constancy of Hodge numbers in the rigid analytic setting,
and deduce that the relative Hodge-Tate spectral sequence degenerates.
