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ABSTRACT Carbon-enhanced metal-poor (CEMP) stars are the living fossils holding records of chemical enrichment from early generations of stars. In this work, we perform a set of numerical simulations of the enrichment from a supernova (SN) of a first generation of metal-free (Pop III) star and the gravitational collapse of the enriched cloud, considering all relevant cooling/heating processes and chemical reactions as well as the growth of dust grains. We adopt faint SN models for the first time with progenitor masses MPopIII = 13–$$80 \ {\rm M_{\bigodot }}$$, which yield C-enhanced abundance patterns ([C/Fe] = 4.57–4.75) through mixing and fallback of innermost layers of the ejecta. This model also considers the formation and destruction of dust grains. We find that the metals ejected by the SN can be partly re-accreted by the same dark matter minihalo, and carbon abundance of the enriched cloud A(C) = 3.80–5.06 is lower than the abundance range of observed CEMP stars (A(C) ≳ 6) because the mass of the metals ejected by faint SNe is smaller than normal core-collapse SNe due to extensive fallback. We also find that cloud fragmentation is induced by gas cooling from carbonaceous grains for $$M_{\rm Pop III}= 13 \ {\rm M_{\bigodot }}$$ even with the lowest iron abundance [Fe/H] ∼ −9. This leads to the formation of low-mass stars, and these ‘giga metal-poor’ stars can survive until the present-day Universe and may be found by future observations.