The controversial ground-state properties of the Kondo insulator Sm B6 have been investigated using B11 NMR in very high magnetic fields up to 37 T. We find evidence that, following the development of a gap in the conduction-band density of states below 100 K, the in-gap states dominate the nuclear relaxation at temperatures less than 10 K. The Korringa product 1/ K2 T1 T exhibits anomalous behavior in this range and the application of high magnetic fields leads to suppression of nuclear relaxation. The hybridization gap, however, remains open up to 37 T. The behavior of the relaxation at low temperatures suggests a strong field dependence of the in-gap states and rules out the possibility that bound states arise from B6 vacancies. A simple density-of-states model and a band scheme are introduced to account for these observations. © 2007 The American Physical Society.