Let $(A,A^+)$ be a sheafy Tate-Huber pair, and let $X=\operatorname{Spa}(A,A^+)$. It is well-known that $H^i(X,\mathcal{O}\_X)=0$ for $i>0$. I assume it is generally not true that $H^i(X,\mathcal{O}\_X^+)=0$ for $i>0$, but I don't think that I have seen an explicit counterexample. Is there a simple example of a nonzero cohomology class in some $H^i(X,\mathcal{O}\_X^+)$ for $i>0$?



Failure of Tate acyclicity for integral structure sheaves
Below is an example of a formal scheme $\mathfrak{X}$ such that its rigid analytic generic fiber $X$ is affinoid, while its special fiber contains a complete elliptic curve $E$. Because $H^1(E,\mathcal{O}\_E) \ne 0$, $H^1(X,\mathcal{O}\_X^+) \ne 0$ as well.


Let $p$ be an odd prime, and let $X$ be the rigid analytic elliptic curve over $\mathbb{Q}\_p$ defined by $$y^2 = x^3-x, \quad |x| \le |p^{-2}|\,.$$
It has a formal model over $\mathfrak{X}$ over $\mathbb{Z}\_p$ that is covered by two affines
$$\mathfrak{U}=\operatorname{Spf}\mathbb{Z}\_p\left<x,y\right>/(y^2-x^3+x)$$
$$\mathfrak{V}=\operatorname{Spf}\mathbb{Z}\_p\left<u,v,w\right>/(v-u^3+uv^2,uw-p)$$
where $u=x/y$, $v=1/y$, $w=py/x$. These correspond to rational subsets $U$ and $V$, defined by the inequalities $|x| \le 1$ and $|y| \ge 1$, respectively.


One can check that the above rings are normal using Serre's criterion. They are also flat over $\mathbb{Z}\_p$. By Theorem 7.4.1 in de Jong's paper [*Crystalline Diedonné module theory via formal and rigid geometry*](http://www.numdam.org/item/PMIHES_1995__82__5_0/), the above rings can be identified with $\mathcal{O}\_X^+(U)$ and $\mathcal{O}\_X^+(V)$.


Now consider $y/x = xu - v \in \mathcal{O}\_X^+(U \cap V)$. I claim that $y/x \notin \mathcal{O}\_X^+(U) + \mathcal{O}\_X^+(V)$. Indeed, consider the elliptic curve defined by $p=0$ on $\mathfrak{U}$ and $w=0$ on $\mathfrak{V}$. On this curve, $y/x$ has simple poles at the points $x=y=0$ and $u=0$, while elements of $\mathcal{O}\_X^+(U) + \mathcal{O}\_X^+(V)$ have either no pole or a pole of order $\ge 2$ at these points. Hence $y/x$ determines a nontrivial element of $H^1(X,\mathcal{O}\_X^+)$.


Since $H^1(X,\mathcal{O}\_X)=0$, the above cocycle must be $p$-power torsion. In fact, it is annihilated by $p$, since $py/x=w$.


