Extending a vertical vector to a vertical vector field

Question

Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, then I know I can extend $D$ to a smooth vector field, say $\tilde{D}$, over all of $M$. But can I always do this in a way so that $\tilde{D}$ is a vertical vector field (i.e., tangent to the fibers of $F$)?