We report the Fermi Large Area Telescope (LAT) detection of two very-high-energy (VHE, E > 100 GeV) gamma -ray photons from the directional vicinity of the distant (redshift, z = 1.1) blazar PKS 0426-380. The null hypothesis that both the 134 and 122 GeV photons originate from unrelated sources can be rejected at the 5.5sigma confidence level. We therefore claim that at least one of the two VHE photons is securely associated with PKS 0426?380, making it the most distant VHE emitter known to date. The results are in agreement with recent Fermi-LAT constraints on the extragalactic background light (EBL) intensity, which imply a z Asymptotically = to 1 horizon for Asymptotically = to100 GeV photons. The LAT detection of the two VHE gamma -rays coincided roughly with flaring states of the source, although we did not find an exact correspondence between the VHE photon arrival times and the flux maxima at lower gamma -ray energies. Modeling the gamma -ray continuum of PKS 0426-380 with daily bins revealed a significant spectral hardening around the time of the first VHE event detection (LAT photon index Gamma Asymptotically = to 1.4) but on the other hand no pronounced spectral changes near the detection time of the second one. This combination implies a rather complex variability pattern of the source in gamma -rays during the flaring epochs. An additional flat component is possibly present above several tens of GeV in the EBL-corrected Fermi-LAT spectrum accumulated over the ~8 month high state.