  The study of top-quark properties will be a central aspect of the physics
programme of any future lepton collider. In this article, we investigate the
production of top-quark pairs in the semi-leptonic decay channel in $\rm
e^+e^-$ collisions, whose experimental signature is one charged lepton, jets,
and missing energy. We present for the first time fiducial cross sections and
differential distributions at next-to-leading-order accuracy in QCD for the
full off-shell process. We find that the QCD corrections for the considered
process are strongly dependent on the beam energies and range from few per cent
up to more than $100\%$ (near threshold and above 1 TeV). We focus, in
particular, on two scenarios: one close to threshold (365 GeV), dominated by
top-pair production, and one at the TeV scale (1.5 TeV), for which
irreducible-background contributions become relevant. An assessment of
polarised-beam effects is also provided.
