The thermal conductivity kappa of the heavy-fermion metal CeCoIn5 was measured in the normal and superconducting states as a function of temperature T and magnetic field H, for a current and field parallel to the [100] direction. Inside the superconducting state, when the field is lower than the upper critical field Hc2, kappa/T is found to increase as T approaches absolute zero, just as in a metal and in contrast to the behavior of all known superconductors. This is due to unpaired electrons on part of the Fermi surface, which dominate the transport above a certain field. The evolution of kappa/T with field reveals that the electron-electron scattering (or transport mass m^*) of those unpaired electrons diverges as H approaches Hc2 from below, in the same way that it does in the normal state as H approaches Hc2 from above. This shows that the unpaired electrons sense the proximity of the field-tuned quantum critical point of CeCoIn5 at H^* = Hc2 even from inside the superconducting state. The fact that the quantum critical scattering of the unpaired electrons is much weaker than the average scattering of all electrons in the normal state reveals a k-space correlation between the strength of pairing and the strength of scattering, pointing to a common mechanism, presumably antiferromagnetic fluctuations.
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