  We report the observation of matter-wave jet emission in a strongly
ferromagnetic spinor Bose-Einstein condensate of $^7$Li atoms. Directional
atomic beams with $|{F=1,m_F=1}\rangle$ and $|{F=1,m_F=-1}\rangle$ spin states
are generated from $|{F=1,m_F=0}\rangle$ state condensates, or vice versa. This
results from collective spin-mixing scattering events, where spontaneously
produced pairs of atoms with opposite momentum facilitates additional
spin-mixing collisions as they pass through the condensates. The matter-wave
jets of different spin states ($|{F=1,m_F=\pm1}\rangle$) can be a macroscopic
Einstein-Podolsky-Rosen state with spacelike separation. Its spin-momentum
correlations are studied by using the angular correlation function for each
spin state. Rotating the spin axis, the inter-spin and intra-spin momentum
correlation peaks display a high contrast oscillation, indicating collective
coherence of the atomic ensembles. We provide numerical calculations that
describe the experimental results at a quantitative level and can identify its
entanglement after 100~ms of a long time-of-flight.
