The k-MENDEL sample of local analogs to reionization galaxies. Spectral identification of EELGs and properties of green peas in DESI
Low-mass galaxies with intense starbursts exhibit spectra dominated by extreme nebular emission and faint stellar continua. These extreme emission-line galaxies (EELGs) are key laboratories to study star formation, feedback, and ionizing photon escape in low-metallicity environments. We exploit the DESI survey to assemble the k-Means of Extreme Nebulae from DEsi outLiers (k-MENDEL), a statistically robust sample of ~16,000 EELGs at 0.01 < z < 0.96 selected via automatic k-means classification. Using SED fitting and Te-based metallicities, we characterize EELGs including "blueberry" and "green pea" galaxies, spanning stellar masses of 10^6-10^10 Msun and SFRs of 0.1-100 Msun/yr. k-MENDEL extends previous SDSS samples toward higher redshifts and lower metallicities (12+log(O/H) ~ 7.0-8.5). EELGs lie systematically above the star-forming main sequence, with sSFRs up to ~100 Gyr^-1. They follow a shallower mass-metallicity relation offset by 0.3-0.5 dex from local relations, closely resembling young galaxies observed with JWST at z > 3-10. The large intrinsic metallicity scatter, even after projecting along the fundamental metallicity relation, indicates strong departures from simple "bathtub" models, suggesting massive inflows of metal-poor gas followed by strong feedback. While ~6% of the sample shows AGN-like signatures, the most extreme star-forming systems reach high ionization (O32 ~ 5-60) comparable to confirmed Lyman-continuum emitters. Our results support the interpretation of EELGs as short-lived, non-equilibrium phases in the evolution of low-mass galaxies and highlight their importance as nearby analogs of galaxies likely driving cosmic reionization (Abridged).