  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 favors 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 $\sim$\,10-20\% 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.
