Here we present high spectral resolution JWST NIRSpec observations of GN 42437, a low-mass (log(M*/M⊙) = 7.9), compact (re < 500pc), extreme starburst galaxy at z = 5.59 with 13 emission line detections. GN 42437 has a low-metallicity (5-10% Z⊙) and its rest-frame Hα equivalent width suggests nearly all of the observed stellar mass formed within the last 3 Myr. GN 42437 has an extraordinary 7σ significant [Ne v] 3427 Å detection. The [Ne v] line has a rest-frame equivalent width of 11 ± 2Å, [Ne v]/Hα = 0.04 ± 0.007, [Ne v]/[Ne iii] 3870Å =0.26 ± 0.04, and [Ne v]/He ii 4687Å =1.2 ± 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 onto 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(MBH/M⊙) ∼ 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.