Formal power series in Berkovich geometry 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.
 A: In Berkovich's paper that you cite, he constructs a $k$-analytic space $\mathfrak{X}_{\eta}$ associated to what he calls a ``special formal $k^{\circ}$-scheme'' $\mathfrak{X}$. Such a formal $k^{\circ}$-scheme is defined to be locally of the form 
\begin{equation}
k^{\circ}\langle T_1,\ldots,T_n\rangle[[S_1,\ldots,S_m]]/I,
\end{equation}
where $I$ is a finitely-presented ideal. This is the space you're interested in, and it's called the analytic generic fibre of $\mathfrak{X}$. 
When there are no formal power series involved (i.e. $\mathfrak{X}$ is locally of the form $k^{\circ}\langle T_1,\ldots,T_n\rangle / I$), we say $\mathfrak{X}$ is topologically of finite presentation over $k^{\circ}$, and $\mathfrak{X}_{\eta}$ is simply (locally) the spectrum
$$
\mathcal{M}(k\langle T_1,\ldots,T_n\rangle / I),
$$
as you guessed. 
However, when there are power series involved, the story becomes a little more complicated. In this case, $\mathfrak{X}_{\eta}$ is (to first approximation) locally the subset of 
$$
\mathcal{M}(k\langle T_1,\ldots,T_n,S_1\ldots,S_m\rangle / I)
$$ 
where $|S_i| < 1$ for all $i=1,\ldots,m$. This explains the example of the product of the closed and open polydiscs that are cited in the question. Of course, the details of this are a little more complicated and are explained in Berkovich's paper, but this is the usual way that one constructs an analytic space associated to a formal power series ring. 
