  We investigate the non-perturbative degrees of freedom of a class of weakly
non-local gravitational theories that have been proposed as an ultraviolet
completion of general relativity. At the perturbative level, it is known that
the degrees of freedom of non-local gravity are the same of the
Einstein--Hilbert theory around any maximally symmetric spacetime. We prove
that, at the non-perturbative level, the degrees of freedom are actually eight
in four dimensions, contrary to what one might guess on the basis of the
"infinite number of derivatives" present in the action. It is shown that six of
these degrees of freedom do not propagate on Minkowski spacetime, but they
might play a role at large scales on curved backgrounds. We also propose a
criterion to select the form factor almost uniquely.
