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Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).

Is the Picard group of $\Omega^{(n)}_K$ known? More generally, I would like to know the Picard group of $\Omega^{(n)}_K$ base changed to any finite extension $L$ of $K$ (which is not the same as $\Omega^{(n)}_L$).

I'd appreciate any references where this question, or similar questions, are considered.

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In this paper of Junger, all these Picard groups are shown to be zero.

I think this was known for a long time for $n=1$, see for example the book Rigid Analytic Geometry and its Applications by Fresnel and van der Put.

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