The emission of $\nu \overline{\nu }$ pairs off electrons in a polarized ultraintense electromagnetic (e.g., laser) wave field is analyzed. We elaborate on the significance of nonlinear electrodynamics effects (i.e., multiphoton processes) and the peculiarities of neutrino production. Special attention is devoted to the convergence of the reaction probabilities as a function of the number of absorbed photons. Expressions for large field intensities are provided. The asymmetry between the probabilities of electron and $\mu +\tau$ neutrino production depends on initial conditions such as energy of the wave field photons and the field intensity. These findings differ from the lowest-order perturbative calculation of the reaction $\gamma +e\to e^{\prime }+\nu \overline{\nu }$.

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