Context. Carbon-enhanced metal-poor (CEMP) stars ([C/Fe] > 0.7) are known to exist in large numbers at low metallicity in the Milky Way halo and are important tracers of early Galactic chemical evolution. However, very few stars of this kind have been identified in the classical dwarf spheroidal (dSph) galaxies, and detailed abundances, including neutron-capture element abundances, have only been reported for 13 stars.Aims. We aim to derive detailed abundances of six CEMP stars identified in the Carina dSph and compare the abundances to CEMP stars in other dSph galaxies and the Milky Way halo. This is the largest sample of CEMP stars in a dSph galaxy analysed to date.Methods. One-dimensional local thermodynamic equilibrium (LTE) elemental abundances are derived via equivalent width and spectral synthesis using high-resolution spectra of the six stars obtained with the MIKE spectrograph at Las Campanas Observatory.Results. We derived abundances or upper limits for up to 27 elements from C to Os in the six stars. Our analysis reveals one of the stars to be a CEMP-no star with very low neutron-capture element abundances. In contrast, the other five stars all show enhancements in neutron-capture elements in addition to their carbon enhancement, classifying them as CEMP-s and -r/s stars. The six stars have similar a and iron-peak element abundances to other stars in Carina, except for the CEMP-no star, which shows enhancement in Na, Mg, and Si. We explored the absolute carbon abundances (A(C)) of CEMP stars in dSph galaxies and find similar behaviour to that seen for Milky Way halo CEMP stars, but highlight that CEMP-r/s stars primarily have very high A(C) values. We also compared the neutron-capture element abundances of the CEMP-r/s stars in our sample to recent i-process yields, which provide a good match to the derived abundances.