We consider nonstandard interactions of neutrinos with electrons arising from a new light spin-1 particle with mass of tens of GeV or lower and couplings to the neutrinos and electron. This boson is not necessarily a gauge boson and is assumed to have no mixing with standard-model gauge bosons. Adopting a model-independent approach, we study constraints on the flavor-conserving and -violating couplings of the boson with the leptons from a number of experimental data. Specifically, we take into account the (anti)neutrino-electron scattering and e(+)e(-) -> nu(nu) over bar gamma measurements and keep explicitly the dependence on the new particle mass in all calculations. We find that one of the two sets of data can provide the stronger constraints, depending on the mass and width of the boson. Also, we evaluate complementary constraints on its separate flavor-conserving couplings to the electron and neutrinos from other latest experimental results.