I suspect that there are examples. Here is an example, relative to the existence of a smooth, projective morphism $\pi:X\to \mathbb{P}^n$ with $\omega_X\otimes \pi^*\mathcal{O}(1)$ ample, for some integer $n\geq 1$. There are examples due to Moret-Bailly of smooth, projective morphisms to $\mathbb{P}^n$ that might do the trick -- I need to double-check the canonical bundle. Anyway, given such a morphism, consider the induced morphism $$ \pi \times \text{Id}_{\mathbb{P}^n}: X\times \mathbb{P}^n \to \mathbb{P}^n\times \mathbb{P}^n.$$ This is also a smooth, projective morphism. Let $d\geq 1$ be an integer such that $pd > n+1$, and consider the closed subscheme, $$ Z = \{ ([x_0,\dots,x_n],[y_0,\dots,y_n])\in \mathbb{P}^n\times \mathbb{P}^n | x_0y_0^{pd} + \dots + x_ny_n^{pd} = 0\}. $$ Define $Y$ to be the inverse image of $Z$ under $\pi\times \text{Id}_{\mathbb{P}^n}$. Of course $Z$ is everywhere smooth. Since $\pi$ is smooth, also $Y$ is smooth. By adjunction, $\omega_Y$ is the restriction to $Y$ of $$ \text{pr}_X^*(\omega_X \otimes \pi^* \mathcal{O}(1)) \otimes \text{pr}_{\mathbb{P}^n}^* \mathcal{O}(dp-n-1), $$ which is ample by hypothesis. On the other hand, there is an action of $\mathbf{\mu}_{pd}^n$ by $$(\lambda_1,\dots,\lambda_n)\cdot ([x_0,\dots,x_n],[y_0,\dots,y_n]) = ([x_0,\dots,x_n],[y_0, \lambda_1y_1,\dots,\lambda_ny_n]).$$