  We present the results of a non-perturbative determination of the improvement
coefficient $c_\mathrm{V}$ and the renormalisation factor $Z_\mathrm{V}$, which
define the renormalised vector current in three-flavour $\mathrm{O}(a)$
improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge
action. In case of the improvement coefficient, we consider both lattice
descriptions of the vector current, the local as well as the conserved (i.e.,
point-split) one. Our improvement and normalisation conditions are based on
massive chiral Ward identities and numerically evaluated in the Schr\"odinger
functional setup, which allows to eliminate finite quark mass effects in a
controlled way. In order to ensure a smooth dependence of the renormalisation
constant and improvement coefficients on the bare gauge coupling, our
computation proceeds along a line of constant physics, covering the typical
range of lattice spacings $0.04\,\mathrm{fm}\lesssim a\lesssim
0.1\,\mathrm{fm}$ that is useful for phenomenological applications. Especially
for the improvement coefficient of the local vector current, we report
significant differences between the one-loop perturbative estimates and our
non-perturbative results.
