The local density of states of a degenerate semiconductor is investigated at low magnetic fields. In order to realize this experiment, we designed a strongly asymmetric double-barrier heterostructure with heavily doped contacts and study the resonant tunneling transport through the lowest localized (zero-dimensional) level in the quantum well. Fine structure in the tunneling current as a function of bias voltage and magnetic field images mesoscopic fluctuations of the local density of states in the emitter contact. Our quantitative analysis demonstrates that the fluctuation pattern originates from quantum interference of diffusive electron waves in the three-dimensional disordered system on the length scale of the mean free path. Since the fluctuations reflect the properties of the electron states below the Fermi level, the observed mesoscopic effect is temperature insensitive.