Lefschetz trace formula for open adic spaces

In this article, we discuss the Lefschetz trace formula for an adic space which is separated smooth of finite type but not necessarily proper over an algebraically closed non-archimedean field. Under a certain condition on the absence of set-theoretical fixed points on the boundary, we obtain a fixed point formula. As an application, we can establish a trace formula for some formal schemes, which is applicable to the Rapoport-Zink tower for GSp(4). A partial generalization of Fujiwara's trace formula for contracting morphisms is also given.