In this Berkovich's paper, the following kind of algebra is studied: $$ A=A_{m,n}=k^\circ \langle T_1,\dots,T_m \rangle [[S_1,\dots,S_n]] $$ where $k$ is some non-archimedean field with non-trivial valuation and $k^\circ$ is the associated ring. If my understanding was right, it is stated (page370, the second last paragraph) that the formal scheme $\mathfrak X=\mathrm{Spf}A$ has generic fiber: $$ \mathfrak X_\eta = E^m(0,1) \times D^n(0,1) $$ where $E^m(0,1)$ and $D^n(0,1)$ are the closed and open polydisks of radius one at $0$.

**Question:**
I hardly encounter (not necessarily convergent) formal power series in the literature. I was wondering if there is a good theory about it. For example, shall we simply consider $\mathcal M(k\langle T_1,\dots, T_m\rangle[[S_1,\dots,S_n]])$ or even $\mathcal M(k[[S_1,\dots,S_n]])$? Are they still analytic spaces?

And, I will really appreciate it if you may mention some reference.