Let $k$ be a complete non-archimedean field with absolute value $| \cdot |$. The Tate algebra over $k$ in the variables $X_1,\ldots,X_n$ is $T_n = k\langle{X_1,\ldots,X_n} \rangle = \{ \sum_{J \geq 0} a_JX^{J}: a_J \in k, |a_J| \to 0 \ \textrm{as} \ J \to \infty \}$, where $J$ a multi-index. An affinoid algebra $R$ is a $k$-algebra for which there exists a surjective morphism $T_n \twoheadrightarrow R$ for some $n$. By a system of topological generators (or an affinoid generating system) $g_1, \ldots, g_r$ of $R$, we mean power bounded elements $g_i \in R$ giving rise to a surjection $T_r \twoheadrightarrow R, X_i \mapsto g_i$. Let $\textrm{Sp}(R)$ be the affinoid space associated to $R$ (whose underlying set is just the spectrum of maximal ideals in $R$). The question:
Given finitely many points $z_1, \ldots, z_s \in \textrm{Sp}(R)$, does there exist an $m \in \mathbb N$ and a system of topological generators $g_1, \ldots, g_m$ of $R$ such that $|g_i(z_j)|<1$ for all $i=1, \ldots, m$ and $j=1, \ldots, s$?
Recall that $g(z) \in R/\mathfrak{m}_z$ is defined as the residue class of $g$ modulo $\mathfrak{m}_z$ and that $R/\mathfrak{m}_z$, being a finite field extension of $k$, is equipped with the unique extension of the absolute value on $k$.
I've tried to figure this out by reducing to the case $R=T_n$. Then each $\mathfrak{m}_{z_i}$ is the kernel of an evaluation map at a point in $c_i \in \overline{k}^n$ of norm $\leq 1$ so modding out $\mathfrak{m}_{z_i}$ is the same as plugging in $c_i$. But I've had no success in solving the problem even in $T_n$.
Help would be much appreciated!