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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.

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I suggest that you look up the Mittag-Leffler decomposition in Fresnel and van der Put's book "Rigid analytic geometry and its applications". It gives you unique representations for elements of the algebra of an affinoid space that is the complement of finitely many open disks in a closed disk (in terms of the sum of a power series in your variable $x$ and of power series in $1/(x-a)$ with one $a$ for each disk you remove).

I am not sure that it is exactly what you are looking for though since it is not explicitly in terms of $P$.

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