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Let $K$ be a number field, let $v$ be a finite place of $K$, let $f\in K[z]$be a degree $d\geqslant2$ polynomial with $v|d$, and let $a\in K$. We show thatif $f$ is postcritically bounded and has potential good reduction with respectto $v$, and if $a$ has a neighborhood in $\mathbb{C}_v$ that is disjoint fromthe postcritical set of $f$, then the arboreal representation associated to thepair $(f,a)$ is either finite or infinitely wildly ramified above $v$. Inparticular, if $f$ is postcritically finite and has potential good reduction at$v$, then the arboreal representation associated to $(f,a)$ is either finite orinfinitely wildly ramified above $v$ for all $a\in K$.

Infinitely wildly ramified arboreal representations for postcritically finite polynomials with potential good reduction