  Using the functional renormalization group, we investigate the electron
instability in the single-sheet BC$_3$ when the electron filling is near a
type-II van Hove singularity. For a finite Hubbard interaction, the
ferromagnetic-like spin density wave order dominates in the immediate vicinity
of the singularity. Elsewhere near the singularity the p-wave superconductivity
prevails. We also find that a small nearest-neighbor Coulomb repulsion can
enhance the superconductivity. Our results show that BC$_3$ would be a
promising candidate to realize topological $p+ip'$ superconductivity, but the
transition temperature is practically sizable only if the local interaction is
moderately strong.
