Motivated by the early-time dynamics of the quark-gluon plasma in high-energy heavy-ion collisions, we extract gluonic spectral functions of overoccupied gauge theories far from equilibrium using classical-statistical lattice simulations and linear response theory. In 3+1 dimensions we find that the spectral function exhibits quasiparticle excitations at all momenta that are mostly consistent with perturbative hard-thermal loop predictions, while partially showing nonperturbative deviations. In contrast, the structure of excitations in 2+1 dimensions is nontrivial and nonperturbative. These nonperturbative interactions lead to broad excitation peaks in the spectral function, demonstrating the absence of soft quasiparticles in these theories. This also suggests that there may be significant nonperturbative corrections present in systems with large momentum anisotropy, which are relevant to phenomenological applications in heavy-ion collisions.