EDIT:  See the comments for why this isn't a good example.  Angelo gives a similar example that actually works.

Suppose that $Y=C$ is a cuspidal curve, and let $\tilde C\to C$ be the normalization.  Put $X = \mathbb{A}^1\times \tilde C$, and let $X\to Y$ be the obvious map.  The fibers are all $\mathbb{A}^1$'s, but over the singularity in $Y$ the map cannot locally be the projection.

If the map $f$ is a submersion and $Y$ is smooth then I think things should work out, at least in characteristic zero.