Branch locus along a smooth curve

Question

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an isomorphism outside $C$. Suppose that $C$ is not contracted by $f$, I stronly feel that $f_{|C}$ is an immersion. Is it true ?