We study spin-transport in bilayer-graphene (BLG), spin-orbit coupled to a tungsten di sulfide (WS$_2$) substrate, and measure a record spin lifetime anisotropy ~40-70, i.e. ratio between the out-of-plane $tau_{perp}$ and in-plane spin relaxation time $tau_{||}$. We control the injection and detection of in-plane and out-of-plane spins via the shape-anisotropy of the ferromagnetic electrodes. We estimate $tau_{perp}$ ~ 1-2 ns via Hanle measurements at high perpendicular magnetic fields and via a new tool we develop: Oblique Spin Valve measurements. Using Hanle spin-precession experiments we find a low $tau_{||}$ ~ 30 ps in the electron-doped regime which only weakly depends on the carrier density in the BLG and conductivity of the underlying WS$_2$, indicating proximity-induced spin-orbit coupling (SOC) in the BLG. Such high $tau_{perp}$ and spin lifetime anisotropy are clear signatures of strong spin-valley coupling for out-of-plane spins in BLG/WS$_2$ systems in the presence of SOC, and unlock the potential of BLG/transition metal dichalcogenide heterostructures for developing future spintronic applications.
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