  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 5.5 sigma confidence level. We therefore claim
that at least one of the two VHE photons is securely associated with the
blazar, making PKS 0426-380 the most distant VHE emitter known to date. The
results are in agreement with the most recent Fermi-LAT constraints on the
Extragalactic Background Light (EBL) intensity, which imply a $z \simeq 1$
horizon for $\simeq$ 100 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 detection of the first VHE event (LAT photon index \Gamma\ $\simeq$
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.
