  We present a detailed analysis of the rest-frame optical emission line ratios
for three spectroscopically confirmed galaxies at $z>7.5$. The galaxies were
identified in the James Webb Space Telescope (JWST) Early Release Observations
field SMACS J0723.3$-$7327. By quantitatively comparing Balmer and oxygen line
ratios of these galaxies with various low-redshift "analogue" populations (e.g.
Green Peas, Blueberries, etc.), we show that no single analogue population
captures the diversity of line ratios of all three galaxies observed at
$z>7.5$. We find that S06355 at $z=7.67$ and S10612 at $z=7.66$ are similar to
local Green Peas and Blueberries. In contrast, S04590 at $z=8.50$ appears to be
significantly different from the other two galaxies, most resembling extremely
low-metallicity systems in the local Universe. Perhaps the most striking
spectral feature in S04590 is the curiously high [O III] $\lambda4363$/[O III]
$\lambda5007$ ratio (RO3) of $0.048$ (or $0.055$ when dust-corrected), implying
either extremely high electron temperatures, $\sim3\times10^4$ K, or gas
densities $>10^4\ {\rm cm^{-3}}$. Observed line ratios indicate that this
galaxy is unlikely to host an AGN. Using photoionization modelling, we show
that the inclusion of high-mass X-ray binaries or a high cosmic ray background
in addition to a young, low-metallicity stellar population can provide the
additional heating necessary to explain the observed high RO3 while remaining
consistent with other observed line ratios. Our models represent a first step
at accurately characterising the dominant sources of photoionization and
heating at very high redshifts, demonstrating that non-thermal processes may
become important as we probe deeper into the Epoch of Reionization.
