Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? In other words, does there exist an analytic adic space $E$ with a morphism to $X$, unique up to isomorphism as an analytic adic space over $X$, such that for every open immersion of analytic adic spaces $S\rightarrow X$ there is a natural isomorphism $\mathcal{E}(S) \cong \mathrm{Hom}_X(S,E)$?

By a theorem of Kedlaya-Liu, we do know that there is a correspondence between $\mathcal{E}$'s and finite projective $A$-modules, which is achieved by mapping $\mathcal{E}$ to the global sections $M=\mathrm{H}^0(X,\mathcal{E})$. So a natural guess for $E$ would be to take $\mathrm{Spa}(B,B^+)$ with $B=\mathrm{Sym}_A(M^\vee)$; however, I do not know what to take for $B^+$. (Is there some sort of $M^+$?)

Thanks!