The HAWC Collaboration has observed gamma rays at energies above 56 TeV from a collection of nine sources. It has been suggested that this emission could be hadronic in nature, requiring that these systems accelerate cosmic-ray protons or nuclei up to PeV-scale energies. In this paper, we instead show that the spectra of these objects favor a leptonic (inverse Compton) origin for their emission. More specifically, the gamma-ray emission from these objects can be straightforwardly accommodated within a model in which similar to O(10%) of the host pulsar's spindown power is transferred into the acceleration of electrons and positrons with a power-law spectrum that extends to several hundred TeV or higher. The spectral break that is observed among these sources is naturally explained within the context of this simple model, and occurs at the energy where the timescale for energy losses matches the age of the pulsar. In contrast, this spectral feature cannot be straightforwardly accommodated in hadronic scenarios. Furthermore, hadronic models predict that these sources should produce more emission at GeV-scale energies than is observed. In light of these considerations, we conclude that HAWC's highest energy sources should be interpreted as TeV halos or pulsar wind nebulae, which produce their emission through inverse Compton scattering, and are powered by the rotational kinetic energy of their host pulsar.