Let $\kappa$ be an algebraically closed field of characteristic 0. Let $X$ be a smooth connected curve defined over $\kappa$, such that there is an unramified cover
$$\pi:X\to \mathbb{A}^1_\kappa$$ Is $\pi$ necessarily an isomorphism?
Of course the assertion is easy if $\kappa=\mathbb{C}$ (by the simply connectedness of $\mathbb{C}$), but is false if $ch(\kappa)=p$: just take the map
$$V(y^p-y-x)\to \mathbb{A}^1_\kappa,\;\; (x,y)\mapsto x$$