Intra-cluster (IC) populations are expected to be a natural result of the hierarchical assembly of clusters, yet their low space densities make them difficult to detect and study. We present the first definitive kinematic detection of an IC population of globular clusters (GCs) in the Virgo cluster, around the central galaxy, M87. This study focuses on the Virgo core for which the combination of NGVS photometry and follow-up spectroscopy allows us to reject foreground star contamination and explore GC kinematics over the full Virgo dynamical range. The GC kinematics changes gradually with galactocentric distance, decreasing in mean velocity and increasing in velocity dispersion, eventually becoming indistinguishable from the kinematics of Virgo dwarf galaxies at $mathrm{R>320, kpc}$. By kinematically tagging M87 halo and intra-cluster GCs we find that 1) the M87 halo has a smaller fraction ($52pm3%$) of blue clusters with respect to the IC counterpart ($77pm10%$), 2) the $(g-r)_{0}$ vs $(i-z)_{0}$ color-color diagrams reveal a galaxy population that is redder than the IC population that may be due to a different composition in chemical abundance and progenitor mass, and 3) the ICGC distribution is shallower and more extended than the M87 GCs, yet still centrally concentrated. The ICGC specific frequency, $S_{N,mathrm{ICL}}=10.2pm4.8$, is consistent with what is observed for the population of quenched, low-mass galaxies within 1~Mpc from the clusters center. The IC population at Virgos center is thus consistent with being an accreted component from low-mass galaxies tidally stripped or disrupted through interactions, with a total mass of $mathrm{M_{ICL,tot}=10.8pm0.1times10^{11}M_{odot}}$.
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