Hi I think that I might have found the solution:
I could extend the derivatives to continuous fuctions on $[0,1]^2$ by defining them via the one sided derivativs along the normal vector.
Now the classical proof of Itô's Lemma (one shows that the result holds true for any polynomial function by using integration by parts formula and kunita-watanabe) should still work, for kunita Watanabe works for continuous functions. Thus Itô's Lemma would be applicable and the problem would be solved.
Could somebody be so kind to give me feedback on this idea (please) :)
Hmm... I might get problems with the 4 corner points - those evil things :(