We have searched for exotic neutrino-electron interactions that could be produced by a neutrino millicharge, by a neutrino magnetic moment, or by dark photons using solar neutrinos in the XMASS-I liquid xenon detector. We observed no significant signals in 711 days of data. We obtain an upper limit for neutrino millicharge of 5.4$times$10$^{-12} e$ at 90% confidence level assuming all three species of neutrino have common millicharge. We also set flavor dependent limits assuming the respective neutrino flavor is the only one carrying a millicharge, $7.3 times 10^{-12} e$ for $ u_e$, $1.1 times 10^{-11} e$ for $ u_{mu}$, and $1.1 times 10^{-11} e$ for $ u_{tau}$. These limits are the most stringent yet obtained from direct measurements. We also obtain an upper limit for the neutrino magnetic moment of 1.8$times$10$^{-10}$ Bohr magnetons. In addition, we obtain upper limits for the coupling constant of dark photons in the $U(1)_{B-L}$ model of 1.3$times$10$^{-6}$ if the dark photon mass is 1$times 10^{-3}$ MeV$/c^{2}$, and 8.8$times$10$^{-5}$ if it is 10 MeV$/c^{2}$.
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