Aims: We present a study of the faint end of the X-ray luminosity function (XLF) of low-mass X-ray binaries (LMXBs) in the Globular Cluster (GC) system of the cD galaxy NGC 1399. Methods: We performed a stacking experiment on 618 X-ray undetected GCs, in order to verify the presence of faint LMXBs and to constrain the faint-end slope of the GC-LMXBs XLF below the individual detection threshold of 8 × 1037 erg s-1 in the 0.5 - 8 keV band. Results: We obtain a significant X-ray detection for the whole GC sample, as well as for the red and blue GC subpopulations, corresponding to an average luminosity per GC ⟨ LX ⟩ GC of (3.6 ± 1.0) × 1036 erg s-1, (6.9 ± 2.1) × 1036 erg s-1, and (1.7 ± 0.9) × 1036 erg s-1, respectively, for all GCs, red GCs, and blue GCs. If LMXBs in red and blue GCs have the same average intrinsic luminosity, we derive a red/blue ratio ≃3 of GCs hosting LMXBs (2.5 ± 1.0 or 4.1 ± 2.5 depending on the surveyed region); alternatively, assuming the fractions observed for brighter sources, we measure an average X-ray luminosity of LX = (4.3 ± 1.3) × 1037 erg s-1 and LX = (3.4 ± 1.7) × 1037 erg s-1 per red and blue GC-LMXBs, respectively. In the assumption that the XLF follows a power-law distribution, we find that a low-luminosity break is required at LX ≤ 8 × 1037 erg s-1 both in the whole, as well as in the color-selected (red and blue) subsamples. Given the bright-end slopes measured above the X-ray completeness limit, this result is significant at >3σ level. Our best estimates for the faint-end slope are βL = -1.39/-1.38/-1.36 for all/red/blue GC-LMXBs. We also find evidence that the luminosity function becomes steeper at luminosities LX ≳ 3 × 1039 erg s-1, as observed in old ellipticals. Conclusions: If most GCs host a single X-ray binary, we conclude that in NGC 1399 the XLF flattens at low luminosities as observed in other nearer galaxies, and we discuss some consequences of this flattening on LMXBs formation scenarios.
