The origin of carbon-enhanced metal-poor (CEMP) stars and their possible connection with the chemical elements produced by the first stellar generation is still highly debated. In contrast to the Galactic halo, not many CEMP stars have been found in the dwarf spheroidal galaxies around the Milky Way. Here we present detailed abundances from ESO VLT/UVES high-resolution spectroscopy for ET0097, the first CEMP star found in the Sculptor dwarf spheroidal. This star has $\text{[Fe/H]}=-2.03\pm0.10$, $\text{[C/Fe]}=0.51\pm0.10$ and $\text{[N/Fe]}=1.18\pm0.20$. The traditional definition of CEMP stars is $\text{[C/Fe]}\geq0.70$, but taking into account that this luminous red giant branch star has undergone mixing, it was intrinsically less nitrogen enhanced and more carbon-rich when it was formed, and so it falls under the definition of CEMP stars, as proposed by Aoki et al. (2007) to account for this effect. By making corrections for this mixing, we conclude that the star had $\text{[C/Fe]}\approx0.8$ during its earlier evolutionary stages. Apart from the enhanced C and N abundances, ET0097 shows no peculiarities in other elements lighter than Zn, and no enhancement of the heavier neutron-capture elements (Ba, La, Ce, Nd, Sm, Eu, Dy), making this a CEMP-no star. However, the star does show signs of the weak $r$-process, with an overabundance of the lighter neutron-capture elements (Sr, Y, Zr). To explain the abundance pattern observed in ET0097, we explore the possibility that this star was enriched by primordial stars. In addition to the detailed abundances for ET0097, we present estimates and upper limits for C abundances in 85 other stars in Sculptor derived from CN molecular lines, including 11 stars with $\text{[Fe/H]}\leq-2$. Combining these limits with observations from the literature, the fraction of CEMP-no stars in Sculptor seems to be significantly lower than in the Galactic halo.