I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$. (On the English Wikipedia, the assumption is smooth, which I suppose means $C^\infty$)

Théorème de Ehresmann

Ehresmann's Lemma

1. Does anybody know of a counterexample in the case where the smoothness is only $C^1$?

2. I am specially interested in the case where the domain of the submersion has dimension 2 and the range dimension 1. I suspect that the theorem holds in this case (select one fiber and build a local fibration-trivialization around it by patching the x-coordinate of local submersion-trivializations where level curves would be horizontals in $\mathbb{R}^2$). Is that already proved or disproved somewhere?