  Relativistic massless Dirac fermions can be probed with high-energy physics
experiments, but appear also as low-energy quasi-particle excitations in
electronic band structures. In condensed matter systems, their massless nature
can be protected by crystal symmetries. Classification of such
symmetry-protected relativistic band degeneracies has been fruitful, although
many of the predicted quasi-particles still await their experimental discovery.
Here we reveal, using angle-resolved photoemission spectroscopy, the existence
of two-dimensional type-II Dirac fermions in the high-temperature
superconductor La$_{1.77}$Sr$_{0.23}$CuO$_4$. The Dirac point, constituting the
crossing of $d_{x^2-y^2}$ and $d_{z^2}$ bands, is found approximately one
electronvolt below the Fermi level ($E_\mathrm{F}$) and is protected by mirror
symmetry. If spin-orbit coupling is considered, the Dirac point degeneracy is
lifted and the bands acquire a topologically non-trivial character. In certain
nickelate systems, band structure calculations suggest that the same type-II
Dirac fermions can be realised near $E_\mathrm{F}$.
