Extremely metal-poor (EMP) stars preserve a fossil record of the composition of the ISM when the Galaxy formed. It is crucial, however, to verify whether internal mixing has modified their surface. We aim to understand the CNO abundance variations found in some, but not all EMP field giants analysed earlier. Mixing beyond the first dredge-up of standard models is required, and its origin needs clarification.The 12C/13C ratio is the most robust diagnostic of deep mixing, because it is insensitive to the adopted stellar parameters and should be uniformly high in near-primordial gas. We have measured 12C and 13C abundances in 35 EMP giants from high-quality VLT/UVES spectra. Correlations with other abundance data are used to study the depth of mixing.The 12C/13C ratio is found to correlate with [C/Fe] (and Li/H), and clearly anti-correlate with [N/Fe]. Evidence for such deep mixing is observed in giants above log L/Lsolar = 2.6, brighter than in less metal-poor stars, but matching the bump in the luminosity function in both cases. Three of the mixed stars are also Na-and Al-rich, another signature of deep mixing, but signatures of the ON cycle are not clearly seen in these stars. Extra mixing processes clearly occur in luminous RGB stars. The Na-and Al-rich giants could be AGB stars themselves, but an inhomogeneous early ISM or pollution from a binary companion remain possible alternatives (abridged).
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