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 $K$).



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> 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 $\Omega^{(n)}\_L$).
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I'd appreciate any references where this question, or similar questions, are considered. 



Picard group of Drinfeld upper half space
In [this](https://arxiv.org/abs/2012.12729) 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.


