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Aims.This study explores the metal enrichment signatures attributed to the first generation of stars (Pop III) in the Universe, focusing on the E-XQR-30 sample – a collection of 42 high signal-to-noise ratio spectra of quasi-stellar objects (QSOs) with emission redshifts ranging from 5.8 to 6.6. We aim to identify traces of Pop III metal enrichment by analyzing neutral gas in the interstellar medium of primordial galaxies and their satellite clumps, detected in absorption. Methods.To chase the chemical signature of Pop III stars, we studied metal absorption systems in the E-XQR-30 sample, selected through the detection of the neutral oxygen absorption line at 1302 Å. The O Iline is a reliable tracer of neutral hydrogen and allowed us to overcome the challenges posed by the Lyman-αforest’s increasing saturation at redshifts above ∼5 to identify damped Lyman-αsystems (DLAs). We detected and analyzed 29 O Isystems atz ≥ 5.4, differentiating between proximate DLAs (PDLAs) and intervening DLAs. Voigt function fits were applied to obtain ionic column densities, and relative chemical abundances were determined for 28 systems. These were then compared with the predictions of theoretical models. Results.Our findings expand the study of O Isystems atz ≥ 5.4 fourfold. No systematic differences were observed in the average chemical abundances between PDLAs and intervening DLAs. The chemical abundances in our sample align with literature systems atz >  4.5, suggesting a similar enrichment pattern for this class of absorption systems. A comparison between these DLA-analogs at 4.5 <  z <  6.5 with a sample of very metal-poor DLAs at 2 <  z <  4.5 shows in general similar average values for the relative abundances, with the exception of [C/O], [Si/Fe] and [Si/O] which are significantly larger for the high-zsample. Furthermore, the dispersion of the measurements significantly increases in the high-redshift bin. This increase is predicted by the theoretical models and indicates a potential retention of Pop III signatures in the probed gas. Conclusions.This work represents a significant advancement in the study of the chemical properties of highly neutral gas atz ≥ 5.4, shedding light on its potential association with the metal enrichment from Pop III stars. Future advancements in observational capabilities, specifically high-resolution spectrographs, are crucial for refining measurements and addressing current limitations in the study of these distant absorption systems.