We report a study of the basal-plane anisotropy of the superfluid density in underdoped YBa2Cu4O8 (Y124), showing the effects of both the CuO2 planes and the fully occupied CuO chains. From small-angle neutron scattering measurements of the vortex lattice, we can infer the superconducting (SC) properties for a temperature (T) range T = 1.5 K to T-c and magnetic induction B from 0.1 to 6 T. We find that the superfluid density along a has a simple d-wave T dependence. However, along b (the chain direction) the superfluid density falls much more rapidly with T and also with increasing field. This strongly suggests the suppression of proximity-effect-induced superconductivity in the CuO chains. In addition, our results do not support a common framework for the low-field in-plane SC response in Y124 and related YBa2Cu3O7, and also indicate that any magnetic field induced charge-density-wave order in Y124 exists only for fields above 6 T.