Suppose I have a morphism f:X→Y such that the relative sheaf of differentials Ω_X/Y is locally free. Does it follow that f is smooth?

The answer is no, but for a silly reason. You could have some non-reducedness (Spec(k[e]/(e^2)) over Spec(k) has a free sheaf of differentials, but isn't smooth). But what if you add the hypothesis that the rank of Ω_X/Y is dim(X)-dim(Y)?

Edit: As Jonathan points out in his answer, I was careless with my counterexample. It only works if char(k)=2.