The combination of low-temperature specific-heat and nuclear-magnetic-resonance (NMR) measurements reveals important information of the ground-state properties of CeRu4Sn6, which has been proposed as a rare example of a tetragonal Kondo-insulator (KI). The NMR spin-latticerelaxation rate 1/T1 deviates from the Korringa law below 100 K signaling the onset of an energy gap ∆Eg1/kB ≈ 30 K. This gap is stable against magnetic fields up to 10 T. Below 10 K, however, unusual low-energy excitations of in-gap states are observed, which depend strongly on the field H. The specific heat C detects these excitations in the form of an enhanced Sommerfeld coefficient γ = C(T )/T : In zero field, γ increases steeply below 5 K, reaching a maximum at 0.1 K, and then saturates at γ ≈ 0.6 J/molK 2 . This maximum is shifted to higher temperatures with increasing field suggesting a residual density of states at the Fermi level developing a spin gap ∆Eg2. A simple model, based on two narrow quasiparticle bands located at the Fermi level -which cross the Fermi level in zero field at 0.022 states/meV f.u. -can account qualitatively as well as quantitatively for the measured observables. In particular, it is demonstrated that fitting our data of both specific heat and NMR to the model, incorporating a Ce magnetic moment of µ = ∆Eg1/µ0H ≈ 1 µB, leads to the prediction of the field dependence of the gap. Our measurements rule out the presence of a quantum critical point as the origin for the enhanced γ in CeRu4Sn6 and suggest that this arises rather from correlated, residual in-gap states at the Fermi level. This work provides a fundamental route for future investigations into the phenomenon of narrow-gap formation in the strongly correlated class of systems.