  We non-perturbatively determine the renormalization factor of the axial
vector current in lattice QCD with $N_f=3$ flavors of Wilson-clover fermions
and the tree-level Symanzik-improved gauge action. The (by now standard)
renormalization condition is derived from the massive axial Ward identity and
it is imposed among Schr\"{o}dinger functional states with large overlap on the
lowest lying hadronic state in the pseudoscalar channel, in order to reduce
kinematically enhanced cutoff effects. We explore a range of couplings relevant
for simulations at lattice spacings of $\approx 0.09$ fm and below. An
interpolation formula for $Z_A(g_0^2)$, smoothly connecting the
non-perturbative values to the 1-loop expression, is provided together with our
final results.
