NGC 3201 is a globular cluster suspected to have an intrinsic spread in the iron content. We re-analyzed a sample of 21 cluster stars observed with UVES-FLAMES at the Very Large Telescope and for which Simmerer et al. (2013ApJ...764L...7S) found a 0.4dex wide [Fe/H] distribution with a metal-poor tail. We confirmed that when spectroscopic gravities are adopted, the derived [Fe/H] distribution spans ~0.4dex. On the other hand, when photometric gravities are used, the metallicity distribution from Fe I lines remains large, while that derived from Fe II lines is narrow and compatible with no iron spread. We demonstrate that the metal-poor component claimed by Simmerer et al. (2013ApJ...764L...7S) is composed by asymptotic giant branch stars that could be affected by non-local thermodynamical equilibrium effects driven by iron overionization. This leads to a decrease of the Fe I abundance, while leaving the Fe II abundance unaltered. A similar finding has been already found in asymptotic giant branch stars of the globular clusters M5 and 47 Tucanae. We conclude that NGC 3201 is a normal cluster, with no evidence of intrinsic iron spread.