Let $(k, |\cdot|)$ be a complete field with a non-Archimedean norm, not necessarily algebraically closed. Define the Tate algebra as follows:

\begin{align*} k \langle T_1, \dots, T_n \rangle = \{ \sum_{m \in \mathbb{N}^n} a_m T_1^{m_1} \cdots T_n^{m_n} \in k[[T_1, \dots, T_n]] \mid \lim_{m \to \infty} a_m = 0 \} \end{align*}

Suppose that $f \in k \langle T_1, \dots, T_n \rangle$ has the property that for every $x \in k^n$ in the closed unitary ball, we have $f(x) = 0$. Does this imply that $f = 0$? I know that this is true if $k$ is algebraically closed because of the maximum principle