Let $p\in\mathbb{N}_{\geq2}$ (not necessarily prime). How can one characterize all $P\in\mathbb{Z}_p[x]$ satisfying $P(\mathbb{Q}_p\setminus\mathbb{Z}_p)\cap\mathbb{Z}_p=\emptyset$ and all $P\in\mathbb{Z}_p[x]$ satisfying $P(\mathbb{Q}_p\setminus\mathbb{Z}_p)\cap p\mathbb{Z}_p=\emptyset$?
I first thought that it would be sufficient to consider only $p$-adic numbers of the form $\frac{n}{q}$ where $n\in\mathbb{Z}_p$ and $q\in\mathbb{P}$ with $q\mid p$ and $q\nmid n$ but $P(x)=4x^2-x$ is an easy example where this is not the case because $P(\frac{1}{4})=0\in\mathbb{Z}_2$ but $P(\frac{2n+1}{2})\notin\mathbb{Z}_2$ for all $n\in\mathbb{Z}$. I suspect that the leading coefficient of the polynomial is important and bounds the values that one has to check somehow (similarily to the rational root theorem for polynomials with integer coefficients).