Topological insulators give rise to exquisite electronic properties due to their spin-momentum locked Dirac-cone-like band structure. Recently, it has been suggested that the required opposite parities between valence and conduction band along with strong spin-orbit coupling can be realized in correlated materials. Particularly, SmB$_6$ has been proposed as candidate material for a topological Kondo insulator. By utilizing scanning tunneling microscopy and spectroscopy measurements down to 0.35 K, we observed several states within the hybridization gap of about $pm$20 meV on well characterized (001) surfaces of SmB$_6$. The spectroscopic response to impurities and magnetic fields allows to distinguish between dominating bulk and surface contributions to these states. The surface contributions develop particularly strongly below about 7 K which can be understood in terms of a breakdown of the Kondo effect at the surface. Our high-resolution data provide detailed insight into the electronic structure of SmB$_6$, which will reconcile many current discrepancies on this compound.
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