Perhaps I've misunderstood the question, but it looks like it's false.

Let M=\{(x,y)∈ℝ²|(x,y)≠(0,0)\}, N=ℝ, and define f(x,y)=x. This is a smooth map of smooth manifolds, with the fibers over ℝ-\{0\} connected, but the fiber over 0 disconnected.