Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
I'm not aware of much. But two works worth noting are:
K.G. Schlesinger. Generalized Manifolds. Chapman & Hall/CRC, 1997.
I.O. Hamad. Generalized curvature and torsion in nonstandard analysis. PhD thesis, Salahaddin University - Erbil, 2007.
Somewhat belatedly we developed foundations for differential geometry using infinitesimal displacements here:
Nowik, T.; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44.