Context. HE 0107−5240 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 HE 0107−5240, suggesting it belongs to a binary system. CEMP-no stars are the closest descendants of the very first Pop III 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 HE 0107−5240 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 4200−4315 Å , 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 HE 0107−5240 emplying spectral synthesis with the SYNTHE and ATLAS codes. Results. Observations of HE 0107−5240 spanning more than 3 years show a monotonic decreasing trend in radial velocity at a rate of approximately by 0.5 m s −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 HE 0107−5240 is a binary star with a long period of about 13000 d (∼ 36 years). The carbon isotopic ratio excludes the possibility that the companion has gone through the AGB phase and transferred mass to the currently observed star. The binarity of HE 0107−5240 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 SMSS 1605−1443.