Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected?

Question

Let $F:\mathbb{R}^n \to \mathbb{R}^m$ be a smooth surjective submersion. Are the fibers of $F$ necessarily connected? What if we substitute $\mathbb{R}^n$ and $\mathbb{R}^m$ with open neighborhoods of the origin?

The problem is motivated by an application of Exercise 8.18 (p.202) of Lee's book on smooth manifolds.