Let $E$ be an elliptic curve with a $p$-torsion point $Q$. Let $X= E \times \mu_p \times \mathbb G_m$. Let $\sigma$ be the automorphism of $X$ that sends $(P,a,b)$ to $(P+Q, a,ab)$. Then $\sigma$ has order $p$ and no fixed points, so the induced map to the quotient $Y=X/\sigma$ is etale.

Let $E$ be an elliptic curve with a $p$-torsion point $Q$. Let $X= E \times (\operatorname{Spec} \mathbb F_p[\epsilon]/\epsilon^2) \times \mathbb G_m$. Let $\sigma$ be the automorphism of $X$ that sends $(P,\epsilon,b)$ to $(P+Q, \epsilon, (1+\epsilon)b)$. Then $\sigma$ has order $p$ and no fixed points, so the induced map to the quotient $Y=X/\sigma$ is etale.

The section $b$ of $\mathbb G_m$ on $Y$ is $\sigma$-invariant when viewed as a section of $\mathbb G_m/\mu_p$. Were $\mathbb G_m / \mu_p$ a sheaf on $X$, it would descend to a section of the presheaf $\mathbb G_m/\mu_p$ on $Y$, which would arise from a section of $\mathbb G_m$ on $Y$, which would lift to a $\sigma$-invariant section of $\mathbb G_m$ on $X$.

But no such section exists, as these would simply be polynomials in $b, b^{-1}$ and $\epsilon$ which mod $\epsilon$ use only $p$-divisible powers of $p$, hence restricted to the closed set $\epsilon=0$, where $\mu_p$ has no nontrivial sections, cannot equal $p$.