Quantum spin liquids (QSLs) are novel phases of matter which remain quantum disordered even at the lowest temperature. They are characterized by emergent gauge fields and fractionalized quasiparticles. Here we show that the sub-kelvin thermal transport of the three-dimensional S=1/2 hyperhyperkagome quantum magnet PbCuTe_{2}O_{6} is governed by a sizeable charge-neutral fermionic contribution which is compatible with the itinerant fractionalized excitations of a spinon Fermi surface. We demonstrate that this hallmark feature of the QSL state is remarkably robust against sample crystallinity, large magnetic field, and field-induced magnetic order, ruling out the imitation of QSL features by extrinsic effects. Our findings thus reveal the characteristic low-energy features of PbCuTe_{2}O_{6} which qualify this compound as a true QSL material.