  We derive chiral Ward identities for lattice QCD with Wilson quarks and
$N_\mathrm{f} \geq 3$ flavours, on small lattices with Schr\"odinger functional
boundary conditions and vanishingly small quark masses. These identities relate
the axial variation of the non-singlet pseudoscalar density to the scalar one,
thus enabling the non-perturbative determination of the scale-independent ratio
$Z_\mathrm{S}/Z_\mathrm{P}$ of the renormalisation parameters of these
operators. We obtain results for $N_\mathrm{f}=3$ QCD with tree-level
Symanzik-improved gluons and Wilson-Clover quarks, for bare gauge couplings
which cover the typical range of large-volume $N_\mathrm{f} = 2+1$ simulations
with Wilson fermions at lattice spacings below $0.1\,$fm. The precision of our
results varies from 0.3\% to 1\%, except for the coarsest lattice, where it is
2\%. We discuss how the $Z_\mathrm{S}/Z_\mathrm{P}$ ratio can be used in the
non-perturbative calculations of $\mathrm{O}(a)$ improved renormalised quark
masses.
