It might be worth pointing out that you get an example of this by localizing at any point of a variety (scheme) which is irreducible but not locally irreducible. In particular, any self-intersecting curve where the intersection is transversal will give an example, just like the one above by Charles.

It might be worth pointing out that you get an example of this by localizing at any point of a variety (scheme) which is irreducible but not (analytically/formally/étale) locally irreducible. In particular, any self-intersecting curve will give an example, just like the one above by Charles.