Irreducibility preserved under étale maps?

Question

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it.

Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic varieties and let $Z \subset Y$ be an irreducible subvariety.

Does it follow that $\pi^{-1}(Z)$ is irreducible? If so, why? If not, do you know a counterexample?

If necessary $X$ and $Y$ can be surfaces over $\mathbb{C}$, the map $\pi$ of degree two, and $Z$ a hyperplane section (i.e. it defines a very ample line bundle).

Thanks!

Edit: I assume the varieties $X$ and $Y$ to be projective.

Answer 1

Hmm...what about $\mathbb{A}^1 - 0 \rightarrow \mathbb{A}^1$ - 0, with $z \mapsto z^2$? Then the preimage of 1 is $\pm 1$, which is not irreducible.