Context. HE0107 − 5240 is a hyper metal-poor star with [Fe / H] = − 5 . 39, one of the lowest-metallicity stars known. Its stellar atmosphere is enhanced in carbon, with [C / Fe] = + 4 . 0, without detectable presence of neutron-capture elements. Therefore, it belongs to the Carbon-Enhanced Metal-Poor (CEMP − no ) group as the majority of most metal-poor stars known to date. Recent studies showed variations in the line-of-sight velocity of HE0107 − 5240, suggesting it belongs to a binary system. CEMP- no stars are the closest descendants of the very first PopIII stars and binarity holds important bearings for the poorly known mechanism leading to their formation. Aims. We performed high-resolution observations with the ESPRESSO spectrograph at the VLT to constrain the kinematical properties of the binary system HE0107 − 5240 and to probe the binarity of the sample of 8 most metal-poor stars with [Fe / H] ≤ − 4 . 5. Methods. Radial velocities are obtained by using cross-correlation in the interval 4200 − 4315Å, which contains the relatively strong CH band, against a template that could be either a synthetic spectrum or a combined observed spectrum in an iterative process. A Bayesian method is applied to calculate the orbit by using the ESPRESSO measurements and others from the literature. Chemical analysis has also been performed in HE0107 − 5240 emplying spectral synthesis with the SYNTHE and ATLAS codes. Results. Observations of HE0107 − 5240 spanning more than 3 years show a monotonic decreasing trend in radial velocity at a rate of ap- proximately by 0.5ms − 1 d − 1 . A maximum v rad was reached between March 13th, 2012, and December 8th, 2014. The period is constrained at P orb = 13009 + 1496 − 1370 d. New more stringent upper limits have been found for several elements: a)[Sr / Fe] and [Ba / Fe] are lower than − 0 . 76 and + 0 . 2 respectively, confirming the star is a CEMP- no ; b)A(Li) < 0 . 5 is well below the plateau at A(Li) = 1 . 1 found in the Lower Red Giant Branch stars, suggesting Li was originally depleted; and c)the isotopic ratio 12 C / 13 C is 87 ± 6 showing very low 13 C in contrast to what expected from a spinstar progenitor. Conclusions. We confirm that HE0107 − 5240 is a binary star with a long period of about 13000d ( ∼ 36years). The carbon isotopic ratio ex-cludes the possibility that the companion has gone through the AGB phase and transferred mass to the currently observed star. The binarity of HE0107 − 5240 implies some of the first generations of low-mass stars form in multiple systems and indicates that the low metallicity does not preclude the formation of binaries. Finally, a solid indication of v rad variation has been found also in SMSS1605 − 1443.