We study how the non-Fermi-liquid nature of the overscreened multi-channel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap ∆. We find that nonFermi liquid features do persist even at finite ∆: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a non-monotonic function of temperature. Even more surprisingly, we also find that the low-energy spectrum in the limit ∆ → 0 actually does not correspond to the spectrum strictly at ∆ = 0. In particular, the ∆ → 0 ground state is an orbitally degenerate spin-singlet, while it is an orbital singlet with a residual spin degeneracy at ∆ = 0. In addition, there are fractionalized spin-1/2 sub-gap excitations whose energy in units of ∆ tends towards a finite and universal value when ∆ → 0; as if the universality of the anomalous power-law exponents that characterise the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by ∆ 6= 0. This intriguing phenomenon can be explained by the renormalisation flow towards the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities.