  Satellite galaxies within the Milky Way's (MW) virial radius
$R_{\mathrm{vir}}$ are typically devoid of cold gas due to ram pressure
stripping by the MW's corona. The density of this corona is poorly constrained
today and essentially unconstrained in the past, but can be estimated using ram
pressure stripping. In this paper, we probe the MW corona at $z\approx 1.6$
using the Draco dwarf spheroidal galaxy. We assume that i) Draco's orbit is
determined by its interaction with the MW, whose dark matter halo we evolve in
time following cosmologically-motivated prescriptions, ii) Draco's star
formation was quenched by ram pressure stripping and iii) the MW's corona is
approximately smooth, spherical and in hydrostatic equilibrium. We used GAIA
proper motions to set the initial conditions and Draco's star formation history
to estimate its past gas content. We found indications that Draco was stripped
of its gas during the first pericentric passage. Using 3D hydrodynamical
simulations at a resolution that enables us to resolve individual supernovae
and assuming no tidal stripping, which we estimate to be a minor effect, we
find a density of the MW corona $\geq 8\times 10^{-4}$ cm$^{-3}$ at a radius
$\approx 0.72R_{\mathrm{vir}}$. This provides evidence that the MW's corona was
already in place at $z\approx 1.6$ and with a higher density than today. If
isothermal, this corona would have contained all the baryons expected by the
cosmological baryon fraction. Extrapolating to today shows good agreement with
literature constraints if feedback has removed $\lesssim 30$% of baryons
accreted onto the halo.
