  The nearby radio galaxy Centaurus A belongs to a class of Active Galaxies
that are very luminous at radio wavelengths. The majority of these galaxies
show collimated relativistic outflows known as jets, that extend over hundreds
of thousands of parsecs for the most powerful sources. Accretion of matter onto
the central super-massive black hole is believed to fuel these jets and power
their emission, with the radio emission being related to the synchrotron
radiation of relativistic electrons in magnetic fields. The origin of the
extended X-ray emission seen in the kiloparsec-scale jets from these sources is
still a matter of debate, although Cen A's X-ray emission has been suggested to
originate in electron synchrotron processes. The other possible explanation is
Inverse Compton (IC) scattering with CMB soft photons. Synchrotron radiation
needs ultra-relativistic electrons ($\sim50$ TeV), and given their short
cooling times, requires some continuous re-acceleration mechanism to be active.
IC scattering, on the other hand, does not require very energetic electrons,
but requires jets that stay highly relativistic on large scales ($\geq$1 Mpc)
and that remain well-aligned with the line of sight. Some recent evidence
disfavours inverse Compton-CMB models, although other evidence seems to be
compatible with them. In principle, the detection of extended gamma-ray
emission, directly probing the presence of ultra-relativistic electrons, could
distinguish between these options, but instruments have hitherto been unable to
resolve the relevant structures. At GeV energies there is also an unusual
spectral hardening in Cen A, whose explanation is unclear. Here we report
observations of Cen A at TeV energies that resolve its large-scale jet. We
interpret the data as evidence for the acceleration of ultra-relativistic
electrons in the jet, and favour the synchrotron explanation for the X-rays.
