Carbon-enhanced metal-poor (CEMP) stars comprise a high percentage of stars at the lowest metallicities. The stars in the CEMP-no subcategory do not show any s-process enhancement and therefore cannot easily be explained by transfer of carbon and s-process elements from a binary AGB companion. We have performed radial velocity monitoring of a sample of 22 CEMP-no stars to further study the role that binarity plays in this type of CEMP star. We find four new binary CEMP-no stars based on their radial velocity variations; this significantly enlarges the population of known binaries to a total of 11. One of the new stars found to be in a binary system is HE 0107–5240, which is one of the most iron-poor stars known. This supports the binary transfer model for the origin of the abundance pattern of this star. We find a difference in binary fraction in our sample that depends on the absolute carbon abundance, with a binary fraction of 47 +15−14% 47 − 14 + 15 % $ 47^{+15}_{-14}\% $ for stars with a higher absolute carbon abundance and 18 +14−9% 18 − 9 + 14 % $ 18^{+14}_{-9}\% $ for stars with a lower absolute carbon abundance. This might imply a relation between a high carbon abundance and the binarity of a metal-poor star. Although binarity does not equate to mass transfer, there is a possibility that a CEMP-no star in a binary system has been polluted, and care has to be taken in the interpretation of their abundance patterns. We furthermore demonstrate the potential of Gaia of discovering additional binary candidates.