We prove that the insulating one-band Hubbard model on the pyrochlore lattice contains, for realistic parameters, an extended quantum spin-liquid phase. This is a three-dimensional spin liquid formed from a highly degenerate manifold of dimer-based states, which is a subset of the classical dimer coverings obeying the ice rules. It possesses spinon excitations, which are both massive and deconfined, and on doping it exhibits spin-charge separation. We demonstrate that the spinons have fermionic statistics, and further that the holons introduced by doping are also fermions. We explain the origin of this counterintuitive result and establish the connection of these emerging fermions with U(1) gauge fields, represented by strings, as anticipated by Levin and Wen.