We calculate the p-wave contribution to the proton-proton fusion S factor and its energy derivative in pionless effective-field theory (EFT) up to next-to-leading order. The leading contributions are given by a recoil piece from the Gamow-Teller and Fermi operators and from relativistic 1/m suppressed weak interaction operators. We obtain the value of (2.5±0.3)×10−28MeVfm2 for the S factor and (2.2±0.2)×10−26fm2 for its energy derivative at threshold. These are smaller than the results of a prior study that employed chiral EFT by several orders of magnitude. We conclude that, contrary to what has been previously reported, the p-wave contribution does not need to be considered in a high-precision determination of the S factor at astrophysical energies. Combined with the chiral EFT calculation of Acharya et al. [Phys. Lett. B 760, 584 (2016)] for the s-wave channel, this gives a total threshold S factor of S(0)=(4.047+0.024−0.032)×10−23MeVfm2.