The introduction of a steady-state spatially periodic Raman coupling between two components of an ultracold atomic gas produces a dressed-state gas with an anisotropic and tunable dispersion relation. A Bose-Einstein condensate (BEC) formed in such a gas is consequently characterized by an anisotropic superfluid critical velocity. The anisotropic dissipation of superfluid flow is quantified by considering the scattering of impurities flowing through this superfluid. A gradual transition from the isotropic nature of an uncoupled BEC to the anisotropic periodically-dressed condensate is obtained as the strength of the Raman coupling is varied. These results present a clear signature for future experimental realizations of this novel superfluid.