  A direct search for charged lepton-flavour violation in top-quark decays is
presented. The data analysed correspond to $79.8\ \text{fb}^{-1}$ of
proton--proton collisions at a centre-of-mass energy of $\sqrt{s}=13\
\text{TeV}$ recorded by the ATLAS experiment at the LHC. The process studied is
the production of top-quark pairs, where one top quark decays into a pair of
opposite-sign different-flavour charged leptons and an up-type quark, while the
other decays semileptonically according to the Standard Model. The signature of
the signal is thus characterised by the presence of three charged leptons, a
light jet and a $b$-jet. A multivariate discriminant is deployed and its
distribution used as input to extract the signal strength. In the absence of a
signal, an upper limit on the branching ratio of ${\cal B}(t \to \ell \ell' q)
< 1.86 \times 10^{-5}$ is set at the 95\% confidence level.
