The $p$-adic norm $$\left|\sum_{n=0}^N a_n x^n\right|= \sup_{n\le N} |a_n|_p$$ is an absolute value on $\Bbb{Z}_p[x]$. The completion, call it $S_1$, is the ring of formal power series whose coefficients $\to 0$, equivalently it is the ring of $p$-adic analytic functions on the closed unit ball of $\Bbb{C}_p$ (this makes sense because it is open).
The difference between analytic functions on the complex unit disk is that here $|f(\zeta)|_p$ is attained for all but finitely many roots of unity
$T_1=S_1[p^{-1}]$ is called "the Tate algebra in one variable" and is a PID.
The fraction field $F_1=Frac(S_1)$ is the ring of meromorphic functions on the closed unit ball. The discrete valuation ring is $$O_{F_1}=\{ f\in F_1,|f|\le 1\}=(S_1-(p))^{-1} S_1$$ The residue field is $O_{F_1}/(p)=\operatorname{Frac}(\Bbb{F}_p[x])$.
From either Krasner's lemma or Weierstrass preparation theorem we should obtain that the poles are algebraic so that $O_{F_1}= \Bbb{Z}_p[x]_{\text{monic}}^{-1}S_1$. Is it correct?
Is there a description of the maximal unramified extension and the tamely ramified extensions of $F_1$? Of the automorphisms of $F_1$? What is the meaning of the extensions and reduction $\bmod p$ in term of analytic functions? Are there more weird absolute values on $\Bbb{Z}_p[x]$? I'll take anything interesting to say about this ring.