We explore the impact of a flavor-dependent intrinsic transverse momentum of quarks on the production of $W^\pm$ bosons in proton-proton collisions at hadron colliders. We build a template fit to the transverse-mass and the lepton transverse-momentum distributions of the $W$ decay products. We estimate the shift in the extracted value of the $W$ mass induced by different choices of flavor-dependent parameters for the quark intrinsic transverse momentum, that all reproduce the same transverse-momentum distribution of the $Z$ boson. Our findings call for more detailed investigations of nonperturbative effects, which might affect very precise determinations of some Standard Model parameters even at kinematics where these effects are expected to be small.