Quantum oscillations in transport and thermodynamic parameters at high magnetic fields are an unambiguous signature of the Fermi surface, the defining characteristic of a metal. Recent observations of quantum oscillations in insulating SmB6 and YbB12, therefore, have been a big surprise—despite the large charge gap inferred from the insulating behaviour of the resistivity, these compounds seemingly host a Fermi surface at high magnetic fields. However, the nature of the ground state in zero field has been little explored. Here, we report the use of low-temperature heat-transport measurements to discover gapless, itinerant, charge-neutral excitations in the ground state of YbB12. At zero field, sizeable linear temperature-dependent terms in the heat capacity and thermal conductivity are clearly resolved in the zero-temperature limit, indicating the presence of gapless fermionic excitations with an itinerant character. Remarkably, linear temperature-dependent thermal conductivity leads to a spectacular violation of the Wiedemann–Franz law: the Lorenz ratio is 104–105 times larger than that expected in conventional metals, indicating that YbB12 is a charge insulator and a thermal metal. Moreover, we find that these fermions couple to magnetic fields, despite their charge neutrality. Our findings expose novel quasiparticles in this unconventional quantum state.