  We give an accurate determination of the vector (electromagnetic) form
factor, $F(Q^2)$, for a light pseudoscalar meson up to squared momentum
transfer $Q^2$ values of 6 $\mathrm{GeV}^2$ for the first time from full
lattice QCD, including $u$, $d$, $s$ and $c$ quarks in the sea at multiple
values of the lattice spacing. Our results show good control of lattice
discretisation and sea quark mass effects. We study a pseudoscalar meson made
of valence $s$ quarks but the qualitative picture obtained applies also to the
$\pi$ meson, relevant to upcoming experiments at Jefferson Lab. We find that
$Q^2F(Q^2)$ becomes flat in the region between $Q^2$ of 2 $\mathrm{GeV}^2$ and
6 $\mathrm{GeV}^2$, with a value well above that of the asymptotic perturbative
QCD expectation, but well below that of the vector-meson dominance pole form
appropriate to low $Q^2$ values. Our calculations show that we can reach higher
$Q^2$ values in future to shed further light on where the perturbative QCD
result emerges.
