This is false in general. Let $X$ be a smooth separated quasicompact rigid space over $\mathbf{Q}_p$. Let $\mathcal{O}^+$ be the sheaf of power-bounded functions. Then the following three statements are actually equivalent:

1) The torsion subgroup of $H^1(X,\mathcal{O}^+)$ is killed by a power of p.

2) For *any* covering $\mathcal{U}$ of $X$ by finitely many affinoids, the torsion subgroup of $\check{H}^{1}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

3) For *some* covering $\mathcal{U}$ of $X$ by finitely many affinoids, the torsion subgroup of $\check{H}^{1}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

This is a straightforward exercise in playing with the Cech-to-derived functor spectral sequence, using critically a theorem of Bartenwerfer that $H^1(U,\mathcal{O}^+)$ is killed by a power of $p$ for any smooth affinoid $U$.

On the other hand, for a fixed covering $\mathcal{U}$ of $X$ by finitely many affinoids and some fixed $n$, consider the following statements.

4) The torsion subgroup of $\check{H}^{n}_{\mathcal{U}}(X,\mathcal{O}^+)$ is killed by a power of $p$.

5) The image of the differential $d^{n-1}_{\mathcal{U}}$ is closed.

Then 5) implies 4), essentially by the open mapping theorem. (It might actually be true that 4) and 5) are equivalent, but I didn't think it through carefully.)

So to find an example where 5) fails (for $n=1$), we just need to find some smooth separated quasicompact $X$ where 1) fails, i.e. where $H^1(X,\mathcal{O}^+)$ contains torsion of arbitrarily high $p$-power order. Such spaces do exist, although not obviously so: de Jong found an example of such an $X$ last year (at my request).

Here is a sketch of de Jong's construction. Take an elliptic curve $E / \mathbf{Z}_p$ with good reduction. Choose a line bundle $L$ on $E$ which is non-torsion in the Picard group but which is trivial modulo $p$. Let $Y \to E$ be the total space of $L$, and let $\mathfrak{Y}$ be the formal completion of $Y$ along $(p)$. Finally, let $X$ be the rigid generic fiber of $\mathfrak{Y}$. Then $H^1(X,\mathcal{O}^+)$ contains torsion of arbitrarily high $p$-power order. (Idea: Reduce to the fact that $H^1(Y,\mathcal{O})=\oplus_{n \geq 0} H^1(E, L^{\otimes n})$ contains torsion of arbitrary $p$-power. Details omitted.)