Does there exist a group-object $Mp^{an}$ in the category of rigid/Berkovich/adic spaces over a non-archimedean field $k$, with a homomorphism $Mp^{an}\to Sp^{an}$ to the analytification of the symplectic group, such that its $k$-points recovers the 'usual' double-covering map of the symplectic group?
My initial thought was to construct such object using an explicit covering of $Sp(k)$, but then the question becomes which covering, and how to glue them in a controllable way? I attempted to create a covering through the cocycles, but that gets messy very quickly.
My pessimistic instinct says that such object might not exist because (1) in the literatures all related metaplectic object are defined on the level of sheaves and not points (quite a meta reason, eh?) (2) there is something that isn't well-behaved on the level of 2-cocycle when we try to base change to another field $k'/k$.
I am looking for a possible construction, or maybe with conditions on $k$ that allow the existence of such object.
