Locally analytic representations in mixed characteristic via stacks

For any $p$-adic Lie group $G$, we construct a group stack $G^\mathrm{la}$ over the $p$-adic branch of Clausen--Scholze's stack of norms such that quasicoherent sheaves on its classifying stack $*/G^\mathrm{la}$ recover locally analytic representations of $G$ after base change to any suitable Tate H