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Mike Anderson wrote a preprint claiming to prove local rigidity of smooth enough surfaces, but it was then later withdrawn.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