We present emission-line ratios from a sample of 27 Lyman-break galaxies from z ∼ 5.5 − 9.5 with −17.0 < M₁₅₀₀ < −20.4, measured from ultra-deep JWST/NIRSpec multi-object spectroscopy from the JWST Advanced Deep Extragalactic Survey (JADES). We used a combination of 28 h deep PRISM/CLEAR and 7 h deep G395M/F290LP observations to measure, or place strong constraints on, ratios of widely studied rest-frame optical emission lines including Hα, Hβ, [O II] λλ3726, 3729, [Ne III] λ3869, [O III] λ4959, [O III] λ5007, [O I] λ6300, [N II] λ6583, and [S II] λλ6716, 6731 in individual z > 5.5 spectra. We find that the emission-line ratios exhibited by these z ∼ 5.5 − 9.5 galaxies occupy clearly distinct regions of line-ratio space compared to typical z ∼ 0 − 3 galaxies, instead being more consistent with extreme populations of lower-redshift galaxies. This is best illustrated by the [O III]/[O II] ratio, tracing interstellar medium (ISM) ionisation, in which we observe more than half of our sample to have [O III]/[O II] > 10. Our high signal-to-noise spectra reveal more than an order of magnitude of scatter in line ratios such as [O II]/Hβ and [O III]/[O II], indicating significant diversity in the ISM conditions within the sample. We find no convincing detections of [N II] λ6583 in our sample, either in individual galaxies, or a stack of all G395M/F290LP spectra. The emission-line ratios observed in our sample are generally consistent with galaxies with extremely high ionisation parameters (log U ∼ −1.5), and a range of metallicities spanning from ∼0.1 × Z_⊙ to higher than ∼0.3 × Z_⊙, suggesting we are probing low-metallicity systems undergoing periods of rapid star formation, driving strong radiation fields. These results highlight the value of deep observations in constraining the properties of individual galaxies, and hence probing diversity within galaxy population.