The first stars were born from chemically pristine gas. They were likely massive, and thus they rapidly exploded as supernovae, enriching the surrounding gas with the first heavy elements. In the Local Group, the chemical signatures of the first stellar population were identified among low-mass, long-lived, very metal-poor ([Fe/H]{<}-2) stars, characterized by high abundances of carbon over iron ([C/Fe]>+0.7): the so-called carbon-enhanced metal-poor stars. Conversely, a similar carbon excess caused by first-star pollution was not found in dense neutral gas traced by absorption systems at different cosmic time. Here we present the detection of 14 very metal-poor, optically thick absorbers at redshift z~3-4. Among these, 3 are carbon-enhanced and reveal an overabundance with respect to Fe of all the analyzed chemical elements (O, Mg, Al, and Si). Their relative abundances show a distribution with respect to [Fe/H] that is in very good agreement with those observed in nearby very metal-poor stars. All the tests we performed support the idea that these C-rich absorbers preserve the chemical yields of the first stars. Our new findings suggest that the first-star signatures can survive in optically thick but relatively diffuse absorbers, which are not sufficiently dense to sustain star formation and hence are not dominated by the chemical products of normal stars.