Let $p$ be a prime number, $\mathbb C_p$ be the completion of an algebraic closure of $\mathbb Q_p$. One denotes $|.|_p$ the norm of $\mathbb C_p$ normalized by $|p|_p=1/p$. Consider a sequence $(f_n)_n$ of power series defined on $\mathcal D=\{z\in\mathbb C_p\mid[z|_p<1\}$. One assumes that $\lim_{n\to+\infty}f_n(z)=1$ for all $z\in\mathcal D$. Can one assert that if $z_0\in\mathcal D$ is a zero (resp a pole) of $F:=\prod_{n=1}^{+\infty}f_n$ then there exist $n_1\in\mathbb N_0$ such that $z_0$ is a zero (resp pole) of $f_{n_1}$.
Thanks in advance for any answer.