The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all holomorphic line bundles are trivial (The exponential exact sequence and the fact that $H^2(X,\mathbb{Z})= 0$). Is the same statement true in the p-adic case? That is, does the vanishing of $H^1(X, \mathcal{O})$ imply the vanishing of $H^1(X, \mathcal{O}^\*)$?



p-adic Stein spaces
The answer is no. In fact, a theorem of Lazard shows that the open unit disk has a non-trivial Picard group when the ground field is a non maximally complete ultrametric valued field.


See Proposition 6 in [Les zéros d'une fonction analytique d'une variable sur un corps valué complet](http://www.numdam.org/item?id=PMIHES_1962__14__47_0). Publications Mathématiques de l'IHÉS, 14 (1962), p. 47-75.


