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Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that issmooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring ofintegers of a $p$-adic field. We construct a fully faithful functor fromperfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfectcomplexes on the v-site of the generic fibre of $X$. Moreover, we describeperfect complexes on the Hodge-Tate stack in terms of certain derivedcategories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adicSimpson functor.

Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces