Let $K$ be a complete, algebraically closed non-Archimedean field, and let $p \in K[x]$ be of degree $d > 0$ and norm 1. (Here the norm of a polynomial is the maximum of the norms of its coefficients.)

I am interested in finding unique representations for elements of the affinoid algebra $$A = K \langle x, p^{-1} \rangle = K \langle x, y \rangle /(p(x)y - 1).$$ Led by the case $p = x$, I thought it might hold that every element of this algebra can be written uniquely as a series $\sum_{i = -\infty}^\infty g_i p^i$, where $g_i \in K[x]$ has $\deg g_i < d$ and $|g_i| \to 0$ as $|i| \to \infty$.

I realised my "proof" of this fact is incorrect, owing to my inability to compare $\left |\sum_{i = -\infty}^\infty g_i p^i \right |$ to $\max_i |g_i|$. It's simple to reduce the claim to elements of $K \langle x \rangle \subseteq A$, and it holds by an easy induction on degree for $K[x]$. But now I'm starting to think the claim itself might be incorrect. If so, is there a standard form of any kind?

If it's relevant, we can assume $p$ has all non-zero coefficients of norm 1 and is divisible by $x$; feel free to add more restrictions to $p$ if they're needed.