Undoped-graphene (Gr) sheets at low temperatures are known, via random-phase-approximation (RPA) calculations, to exhibit unusual van derWaals (vdW) forces. Here, we show that graphene is the first known system where effects beyond the RPA within each interacting subsystem make qualitative changes to the vdW force, observable via its local exponent d logƞ=d logĞ. For large separations D ? 10 nm, where only the pz vdW forces remain, we find that the Gr-Gr vdW interaction is substantially reduced from the RPA prediction. Its D dependence is very sensitive to the form of the long-wavelength, in-plane many-body enhancement of the velocity of the massless Dirac fermions and may provide independent confirmation of the latter via direct force measurements. The simple connection that we expose is a strong motivation for further refinement of recent successful direct vdW force easurements.