  Nonperturbative renormalization group techniques have recently proven a
powerful tool to tackle the nontrivial infrared dynamics of light scalar fields
in de Sitter space. In the present article, we develop the formalism beyond the
local potential approximation employed in earlier works. In particular, we
consider the derivative expansion, a systematic expansion in powers of field
derivatives, appropriate for long wavelength modes, that we generalize to the
relevant case of a curved metric with Lorentzian signature. The method is
illustrated with a detailed discussion of the so-called local potential
approximation prime which, on top of the full effective potential, includes a
running (but field-independent) field renormalization. We explicitly compute
the associated anomalous dimension for O(N) theories. We find that it can take
large values along the flow, leading to sizable differences as compared to the
local potential approximation. However, it does not prevent the phenomenon of
gravitationally induced dimensional reduction pointed out in previous studies.
We show that, as a consequence, the effective potential at the end of the flow
is unchanged as compared to the local potential approximation, the main effect
of the running anomalous dimension being merely to slow down the flow.
