TITLE: Reference Request: Vector bundles in rigid analytic geometry
QUESTION [7 upvotes]: In algebraic geometry it is well-known (see Hartshorne Exercise II.5.16 for example) that there is a 1-1 correspondence between rank $n$ (geometric) vector bundles $\pi\colon Y\to X$ on a scheme $X$ and locally free sheaves of $\mathcal{O}_X$-modules of rank $n$.
As I would imagine is well-known to experts in the subject, there is an analogous result in rigid analytic geometry (in the sense of Tate). Here the analogue of a trivial geometric vector bundle of rank $n$ on a rigid $K$-analytic space $X$ is the fibre product $X\times_K \mathbb{A}^{n,an}$ together with its natural projection onto $X$. Here $\mathbb{A}^{n,an}$ denotes the rigid analytic space obtained by gluing together polydiscs of larger and larger radius as in Example 9.3.4.1 of Non-Archimedean analysis by Bosch, Güntzer and Remmert. One can verify that the sections of the projection map $X\times_K \mathbb{A}^{n,an}\to X$ are naturally a free $\mathcal{O}_X(X)$-module of rank $n$. Given this it is not difficult to make a definition of a general geometric vector bundle on rank $n$ on a rigid $K$-analytic space in such a way that the sections of such a bundle form a locally free sheaf of $\mathcal{O}_X$-modules (that is a vector bundle on $X$ in the only sense I can find in the literature). Moreover, as in the algebraic setting this actually defines a 1-1 correspondence between the two notions of vector bundle. 
My question is whether anyone has written this up in a form that can be easily cited.

REPLY [2 votes]: My guess is that in chapter 4, specially section 4.7, of 
Rigid Analytic Geometry and Its Applications, by Jean Fresnel and Marius Van Der Put, Progress in Mathematics, vol. 218, Birkhäuser, Boston, 2004.
you could find the result you are looking for.