In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero section such that all stalks vanish (at all the "usual" points corresponding to maximal ideals in the affinoid algebra). He uses this to motivate Berkovich spaces etc, and explains why the existence of such a section does not contradict anything (the resulting open cover on which the section vanishes might not be an admissible cover) but does not give an explicit example of such an affinoid/sheaf/section.

What is an explicit example?