Context. Carbon-enhanced metal-poor (CEMP) stars are common objects in the metal-poor regime. The lower the metallicity we look at, the larger the fraction of CEMP stars with respect to metal-poor stars with no enhancement in carbon. The chemical pattern of CEMP stars is diversified, strongly suggesting a different origin of the C enhancement in the different types of CEMP stars. Aims. We selected a CEMP star, SDSS J0222-0313, with a known high carbon abundance and, from a low-resolution analysis, a strong enhancement in neutron-capture elements of the first peak (Sr and Y) and of the second peak (Ba). The peculiarity of this object is a greater overabundance (with respect to iron) of the first s-process peak than the second s-process peak. Methods. We analysed a high-resolution spectrum obtained with the Mike spectrograph at the Clay Magellan 6.5 m telescope in order to derive the detailed chemical composition of this star. Results. We confirmed the chemical pattern we expected; we derived abundances for a total of 18 elements and significant upper limits. Conclusions. We conclude that this star is a carbon-enhanced metal-poor star enriched in elements produced by s-process (CEMP-s), whose enhancement in heavy elements is due to mass transfer from the more evolved companion in its asymptotic giant branch (AGB) phase. The abundances imply that the evolved companion had a low main sequence mass and it suggests that it experienced a proton ingestion episode at the beginning of its AGB phase.