Résumé : A large fraction of the carbon-enhanced, extremely metal-poor halo giants ([Fe/H] <-2.5) are also strongly enriched in neutron-capture elements from the s process (CEMP-s stars). The conventional explanation for the properties of these stars is mass transfer from a nearby binary companion on the asymptotic giant branch (AGB). This scenario leads to a number of testable predictions in terms of the properties of the putative binary system and the resulting abundance pattern. Among the CEMP stars, some stars further exhibit overabundances in r-process elements on top of the s-process enrichment, and are tagged CEMP-rs stars. Although the nucleosynthesis process responsible for this kind of mixed abundance pattern is still under debate, CEMP-rs stars seem to belong to binary systems as do CEMP-s stars. Aims. Our aim is to present and analyse in detail our comprehensive data set of systematic radial-velocity measurements and high-resolution spectroscopy of the CEMP star HE 0017+0055. Methods. Our precise radial-velocity monitoring of HE 0017+0055 over 2940 days (8 yr) with the Nordic Optical Telescope and Mercator telescopes exhibits variability, with a period of 384 d and amplitude of 540 ± 27 m s-1 superimposed on a nearly linear long-term decline of ~1 m s-1 day-1. We used high-resolution HERMES/Mercator and Keck/HIRES spectra to derive elemental abundances with 1D LTE MARCS models. A metallicity of [Fe/H] ~-2.4 is found, along with s-process overabundances of the order of 2 dex (with the exception of [Y/Fe] ~ +0.5), and most notably overabundances of r-process elements like Sm, Eu, Dy, and Er in the range 0.9-2.0 dex. With [Ba/Fe] > 1.9 dex and [Eu/Fe] = 2.3 dex, HE 0017+0055 is a CEMP-rs star. We used the derived atmospheric parameters and abundances to infer HE 0017+0055 evolutionary status from a comparison with evolutionary tracks. Results. HE 0017+0055 appears to be a giant star below the tip of the red giant branch. The s-process pollution must therefore originate from mass transfer from a companion formerly on the AGB, which is now a carbon-oxygen white dwarf (WD). If the 384 d velocity variations are attributed to the WD companion, its orbit must be seen almost face-on, with i ~ 2.3°, because the mass function is very small: f(M1,M2) = (6.1 ± 1.1) × 10-6M⊙. Alternatively, the WD orbital motion could be responsible for the long-term velocity variations, with a period of several decades. The 384 d variations should then be attributed either to a low-mass inner companion (perhaps a brown dwarf, depending on the orbital inclination), or to stellar pulsations. The latter possibility is made likely by the fact that similar low-amplitude velocity variations, with periods close to 1 yr, have been reported for other CEMP stars in a companion paper. A definite conclusion about the origin of the 384 d velocity variations should however await the detection of synchronous low-amplitude photometric variations.