Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H?
Commentary: I realise that I am not being overly precise here. This is partly because I am not sure in exactly what form I want to ask this question. In lieu of this, I will mention an example, that I suspect may produce an interesting quotient space.
Let Spec(A) be an affine group scheme of finite type over ko. Let G be the analytification of the generic fibre of Spec(A). On $A\otimes_{k^\circ} k$, one can define the seminorm where ||a|| is defined to be the infimum of all |t| where $t\in k^\times$ is such that $a\in tA=t(A\otimes_{k^\circ} 1)\subset A\otimes_{k^\circ} k$. Take the completion of this tensor product with respect to this seminorm, and let H be its spectrum (which is then naturally an affinoid subgroup of G).
Best Answer
There is an old and forgotten paper of Z. Bosch on homogeneous spaces for affinoid groups (in German), where he construct rigid analytic quotients for affinoid groups and also proves some properties. The same arguments should work in the case of Berkovich spaces.