We report on the direct observation of resonant electric dipole-dipole
interactions in a cubic array of atoms in the many-excitation limit. The
interactions, mediated by single-atom couplings to the shared electromagnetic
vacuum, are shown to produce spatially-dependent cooperative Lamb shifts when
spectroscopically interrogating the mHz-wide optical clock transition in
strontium-87. We show that the ensemble-averaged shifts can be suppressed below
the level of evaluated systematic uncertainties for state-of-the-art optical
atomic clocks. Additionally, we demonstrate that excitation of the atomic
dipoles near a Bragg angle can enhance these effects by nearly an order of
magnitude compared to non-resonant geometries. Given the remarkable precision
of frequency measurements and the high accuracy of the modeled response, our
work demonstrates that such a clock is a novel platform for studies of the
quantum many-body physics of spins with long-range interactions mediated by
propagating photons