We present a search for galaxies in the local Universe with extremely low oxygen abundance, that is, more than 25 times lower than solar, which corresponds to 12 + log(O/H) < 7.3. To determine the oxygen abundance, we apply the direct Te method for objects where the [O III]λ4363 line is detected. We identified 21 extremely metal-poor galaxies in the early data release of the Dark Energy Spectroscopic Instrument (DESI EDR), for some of which we also derived N/O, Ne/O, Ar/O, and S/O ratios. We find that many DESI galaxies with extremely low oxygen abundance exhibit a higher N/O ratio in comparison to the reference low-metallicity sample collected from the literature. We suggest that the elevation in N/O ratio may be explained by a contamination with metal-rich gas caused by gas inflow or a merger event. Moreover, contrary to some recent studies, we find that Ar/O and S/O ratios are enhanced as well, while the Ne/O ratio does not show such elevation. One of the galaxies, J0713+5608, has a remarkably low oxygen abundance of 6.978 ± 0.095 dex. This measurement aligns with the lowest known oxygen abundances in galaxies to date. Given the relatively high uncertainty, this galaxy may have the lowest oxygen abundance ever found. Additionally, J0713+5608 exhibited an enhanced N/O ratio compared to the typical N/O ratio observed in metal-poor galaxies within the local Universe.