Trivialising cover for étale morphisms

Question

Let $f:Y \to X$ be a finite étale morphism of smooth and proper schemes over a field $k$ (not necessarily separable closed).

Is there a geometrically connected étale cover $\{U_i\}$ of $X$ which trivializes $Y$, i.e. $Y \times_k U_i$ is the disjoint union of copies of $U_i$?

EDIT: it seems that, for general $k$, the above is not necessarily true. I have a couple of follow-up questions:

• If $k = k^{sep}$, is there a positive answer to my original question?

• Is there a connected étale cover $\{U_i\}$ of $X$ which trivializes $Y$, for a general $k$?