Context. The vast majority of the most iron-poor stars in the Galaxy exhibit strong carbon enhancement, with C/H ratios about two orders of magnitude below solar. This unusual chemical composition likely reflects the properties of the gas cloud from which these stars formed, having been enriched by one, or at most a few, supernovae. A remarkable member of this stellar class, HE 0107 −5240 with [Fe/H]=─5.56, has been identified as part of a binary system. Aims. We aim to constrain the orbital parameters of HE 0107−5240 through radial velocity monitoring with the ESPRESSO spectrograph. Methods. We derived radial velocities using cross-correlation with a template, taking advantage of the strong G-band feature. We combined all observations to obtain a high signal-to-noise ratio spectrum, which we used to refine our understanding of the stellar chemical composition. We additionally used a co-added UVES spectrum in the blue to complement the wavelength coverage of ESPRESSO. Results. Observations of HE 0107−5240 over a span of more than four years yield a revised orbital period of about 29 years. We provide updated elemental abundances for Sc, Cr, Co, and, tentatively, Al, along with a new upper limit for Be. We derive the iron abundance from ionised Fe lines and establish significant upper limits for Li, Si, and Sr. Conclusions. We confirm that the star is a long-period binary. Iron abundances derived from neutral and ionised lines are consistent with the local thermodynamical equilibrium (LTE) assumption, which casts doubt on published deviations from LTE corrections for Fe in this star. The heavy elements Sr and Ba remain undetected, confirming the classification of HE 0107−5240 as a carbon-enhanced, metal-poor star not enhanced in heavy elements (CEMP-no), and supporting the absence of an n-capture element plateau at the lowest metallicities.