This paper focuses on robust stability and performance of uncertain nonlinear systems with structured locally bounded uncertainty. It is shown that, in a general setting under a considerably mild structural assumption, a local small gain condition implies robust local performance. The feature of this paper is that the performance is defined in a local sense, which is a realistic property inherently nonlinear systems can achieve. The local robustness is analyzed purely in terms of input-output properties so that the proof involves neither state-space descriptions nor Hamilton-Jacobi equations.