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