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We present the detection and analysis of H2 absorption at z = 4.24 towards the bright quasar J 0007−5705, which was observed with the Very Large Telescope as part of the ESPRESSO QUasar Absorption Line Survey (EQUALS). The high resolving power of R ≈ 120 000 enables the identification of extremely weak H2 lines in several rotational levels at a total column density of N(H2)≈2 × 1014 cm−2, which is among the lowest ever measured in quasar absorption systems. Remarkably, this constitutes the highest redshift H2 detection to date. Two velocity components are resolved that are separated by only 3 km s−1: a narrow (b ∼ 1.7 km s−1) and a broader (b ≃ 6.2 km s−1) component. Modelling the rotational population of H2 yields a density of log nH/cm−3 ∼ 2.8 and temperature of ∼40 K (typical of the cold neutral medium) for the narrow component and log nH/cm−3 ∼ 1.4, T ∼ 600 K for the warmer, more turbulent component under a moderate ultraviolet (UV) field, suggesting at least a several-megaparsec distance from the quasar. This system reveals the existence of tiny (down to ∼0.01 pc), cold overdensities in the neutral medium. Their detection among only seven damped Lyman-α systems in EQUALS suggests that they may be widespread yet usually remain undetected. H2 provides an exceptionally sensitive probe of these structures: even a minute molecular fraction produces measurable Lyman-Werner absorption lines along the extremely narrow optical beam –the size of the quasar’s accretion disc– when observed at sufficiently high spectral resolution. High-resolution spectroscopy on extremely large telescopes may routinely detect and resolve such structures in the distant Universe, where 21-cm absorption traces the collective contribution of many cold cloudlets towards larger radio background sources.