  We report on measurements of quantized conductance in gate-defined quantum
point contacts in bilayer graphene that allow the observation of subband
splittings due to spin-orbit coupling. The size of this splitting can be tuned
from 40 to 80 $\mu$eV by the displacement field. We assign this gate-tunable
subband-splitting to a gap induced by spin-orbit coupling of Kane-Mele type,
enhanced by proximity effects due to the substrate. We show that this
spin-orbit coupling gives rise to a complex pattern in low perpendicular
magnetic fields, increasing the Zeeman splitting in one valley and suppressing
it in the other one. In addition, we observe the existence of a spin-polarized
channel of 6 e$^2$/h at high in-plane magnetic field and of signatures of
interaction effects at the crossings of spin-split subbands of opposite spins
at finite magnetic field.
