  We investigate the effects of ram pressure on the ordered magnetic field of a
galaxy hosting a radio halo and strong nuclear outflows. New radio images in
total and polarized intensity of the edge-on Virgo galaxy NGC\,4388 were
obtained within the CHANG-ES EVLA project. The unprecedented noise level
reached allows us to detect striking new features of the ordered magnetic
field. The nuclear outflow extends far into the halo to about 5\,kpc from the
center and is spatially correlated with the $\rm{H}\alpha$ and X-ray emission.
For the first time, the southern outflow is detected. Above and below both
spiral arms we find extended blobs of polarized emission with an ordered field
oriented perpendicular to the disk. The synchrotron lifetime of the cosmic ray
electrons (CREs) in these regions yields a mean outflow velocity of
$(270\pm70)\kms$, in agreement with a galactic wind scenario. The observed
symmetry of the polarized halo features in NGC 4388 excludes a compression of
the halo gas by the ram pressure of the intra-cluster medium (ICM). The
assumption of equilibrium between the halo pressure and the ICM ram pressure
yields an estimate of the ICM density that is consistent with both the ICM
density derived from X-ray observations and the recent \textit{Planck}
Sunyaev-Zel'dovich measurements. The detection of a faint radio halo around
cluster galaxies could thus be used for an estimate of ICM ram pressure.
