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The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question.

In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is generalized from holomorphic functions to continuously differentiable functions from $\mathbb{R}^2 \rightarrow \mathbb{R}^2$:

$f(z) = \frac{1}{2\pi i} \oint_{\partial \Omega} \frac{f(\zeta)}{\zeta-z} d\zeta - \frac{1}{2\pi i} \iint_\Omega \frac{\partial f/\partial \overline{\zeta} }{\zeta-z} d\overline{\zeta} \wedge d\zeta $

Does this integral formula imply any rigidity for functions that are only continuously differentiable ($C^1$) but 2-dimensional ?

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That theorem of Krantz's is basically just a reformulation of Green's theorem. The only hope you have for having "analytic-like rigidity" coming from Green's theorem would be to weaken your notion of "analytic-like" to the point that it has no resemblance to "analytic". –  Ryan Budney Apr 22 '12 at 22:34
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What exactly do you mean by rigidity? –  Igor Rivin Apr 22 '12 at 23:32
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