ABSTRACT Here, we present high spectral resolution ${\it JWST}$ NIRSpec observations of GN 42437, a low-mass (log(M$_\ast /{\rm M}_\odot)=7.9$), compact ($r_e \lt 500$pc), extreme starburst galaxy at $z=5.59$ with 13 emission-line detections. GN 42437 has a low metallicity (5–10 per cent Z$_\odot$) and its rest-frame H$\alpha$ equivalent width suggests nearly all of the observed stellar mass formed within the last 3 Myr. GN 42437 has an extraordinary 7$\sigma$ significant [Ne v] 3427 Å detection. The [Ne v] line has a rest-frame equivalent width of $11\pm 2$Å, [Ne v]/H$\alpha =0.04\pm 0.007$, [Ne v]/[Ne iii] 3870Å $= 0.26\pm 0.04$, and [Ne v]/He ii 4687Å $ = 1.2\pm 0.5$. Ionization from massive stars, shocks, or high-mass X-ray binaries cannot simultaneously produce these [Ne v] and other low-ionization line ratios. Reproducing the complete nebular structure requires both massive stars and accretion on to a black hole. We do not detect broad lines nor do the traditional diagnostics indicate that GN 42437 has an accreting black hole. Thus, the very high-ionization emission lines powerfully diagnose faint narrow-line black holes at high redshift. We approximate the black hole mass in a variety of ways as log(M$_{\rm BH}/{\rm M}_\odot) \sim 5{-}7$. This black hole mass is consistent with local relations between the black hole mass and the observed velocity dispersion, but significantly more massive than the stellar mass would predict. Very high-ionization emission lines may reveal samples to probe the formation and growth of the first black holes in the universe.
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