Maximum modulus principle over the $p$-adic integers

Question

Consider $\mathbb{Z}_p$ the $p$-adic integers. Let $f\in\mathbb{Z}_p[x]$ be an arbitrary polynomial in one variable. Write $f(x) = \sum_{k}a_kx^k$. Is it true that $\|f\|:= \max_k |a_k|_p = \sup_{t \in \mathbb{Z}_p}|f(t)|_p$?

I know that over a non-archimedean field with infinite residue field this works. My hope is that this works over $\mathbb{Z}_p$ for polynomials in one variable.

Answer 1

No. Let $K$ be a nonarchimedean discrete valued field with a finite residue of order $q$. Let $f(x) = x^q - x$ and $\pi$ be a uniformizer in $K$ (generator of $\mathfrak m_K$). Then $|\!|f|\!| := \max_k |a_k| = 1$ but $\sup_{t \in \mathcal O_K} |f(t)| \leq |\pi| < 1$.

So when the residue field is finite and the valuation is discrete there is always a counterexample.