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Apparently the question is still open for smooth enough surfaces and deformations (that is, at least $C^2$).

Mike Anderson wrote a preprint claiming to prove local rigidity of smooth enough surfaces, but it was then later withdrawn.

Idjad Sabitov and his collaborators have been working on this question, developing for instance a theory of higher-order isometric deformations, see e.g. Sabitov, I. Kh. Local theory of bendings of surfaces [MR1039820 (91c:53004)]. Geometry, III, 179–256, Encyclopaedia Math. Sci., 48, Springer, Berlin, 1992. He conjectures that local rigidity holds for analytic surfaces.

Apparently the question is still open for smooth enough surfaces and deformations (that is, at least $C^2$).