= Solution
Properness is essential. The projection
$$
F:\mathbb R^2\setminus\{(0,0)\}\longrightarrow\mathbb R,
\qquad F(x,y)=x,
$$
is a submersion but is not proper. Its fiber over $x\ne0$ is diffeomorphic to $\mathbb R$, whereas its fiber over zero is $\mathbb R\setminus\{0\}$ and has two connected components.
Merely requiring every fiber to be a submanifold is also insufficient. The surjective map
$$
f:\mathbb R\longrightarrow\mathbb R,\qquad f(x)=x^3-x,
$$
has every fiber finite and therefore a zero-dimensional <embedded submanifold>. Some regular values have three preimages and others have one, so the fibers are not all diffeomorphic. The map fails to be a submersion at $x=\pm1/\sqrt3$.
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