We report the first observation of a large spin-lifetime anisotropy in bilayer graphene (BLG) fully encapsulated between hexagonal boron nitride. We characterize the out-of-plane (tau(perpendicular to)) and in-plane (tau(parallel to)) spin lifetimes by oblique Hanle spin precession. At 75 K and the charge neutrality point (CNP), we observe a strong anisotropy of tau(perpendicular to)/tau(parallel to) = 8 +/- 2. This value is comparable to graphene-transition-metal-dichalcogenide heterostructures, whereas our high-quality BLG provides with tau(perpendicular to) up to 9 ns, a spin lifetime more than 2 orders of magnitude larger. The anisotropy decreases to 3.5 +/- 1 at a carrier density of n = 6 x 10(11) cm(-2). Temperature-dependent measurements show above 75 K a decrease of tau(perpendicular to)/tau(parallel to) with increasing temperature, reaching the isotropic case close to room temperature. We explain our findings with electric-field-induced spin-valley coupling arising from the small intrinsic spin-orbit fields in BLG of 12 mu eV at the CNP.