  We present a spectroscopic study of 287 Planetary Nebulas (PNs) in a total
area of ~0.4 deg^2 around the BCG M87 in Virgo A. With these data we can
distinguish the stellar halo from the co-spatial intracluster light (ICL). PNs
were identified from their narrow and symmetric redshifted lambda 5007\4959
Angstrom [OIII] emission lines, and the absence of significant continuum. We
implement a robust technique to measure the halo velocity dispersion from the
projected phase-space to identify PNs associated with the M87 halo and ICL. The
velocity distribution of the spectroscopically confirmed PNs is bimodal,
containing a narrow component centred on the systemic velocity of the BCG and
an off-centred broader component, that we identify as halo and ICL,
respectively. Halo and ICPN have different spatial distributions: the halo PNs
follow the galaxy's light, whereas the ICPNs are characterised by a shallower
power-law profile. The composite PN number density profile shows the
superposition of different PN populations associated with the M87 halo and the
ICL, characterised by different PN alpha-parameters, the ICL contributing ~3
times more PNs per unit light. Down to m_5007=28.8, the M87 halo PN luminosity
function (PNLF) has a steeper slope towards faint magnitudes than the IC PNLF,
and both are steeper than the standard PNLF for the M31 bulge. Moreover, the IC
PNLF has a dip at ~1-1.5 mag fainter than the bright cutoff, reminiscent of the
PNLFs of systems with extended star formation history. The M87 halo and the
Virgo ICL are dynamically distinct components with different density profiles
and velocity distribution. The different alpha values and PNLF shapes of the
halo and ICL indicate distinct parent stellar populations, consistent with the
existence of a gradient towards bluer colours at large radii. These results
reflect the hierarchical build-up of the Virgo cluster.
