We present emission line ratios from a sample of 26 Lyman break galaxies from z ∼ 5.5 − 9.5 with −17.0 < M 1500 < −20.4, measured from ultra-deep JWST/NIRSpec MSA spectroscopy from the JWST Advanced Deep Extragalactic Survey (JADES). We use a combination of 28 hour deep PRISM/CLEAR and 7 hour deep G395M/F290LP observations to measure, or place strong constraints on, ratios of widely studied rest-frame optical emission lines including Hα, Hβ,, 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.