  The $L_{\mu}-L_{\tau}$ extension of the standard model predicts the existence
of a lepton-flavor-universality-violating $Z^{\prime}$ boson that couples only
to the heavier lepton families. We search for such a $Z^\prime$ through its
invisible decay in the process $e^+ e^- \to \mu^+ \mu^- Z^{\prime}$. We use a
sample of electron-positron collisions at a center-of-mass energy of 10.58GeV
collected by the Belle II experiment in 2019-2020, corresponding to an
integrated luminosity of 79.7fb$^{-1}$. We find no excess over the expected
standard-model background. We set 90$\%$-confidence-level upper limits on the
cross section for this process as well as on the coupling of the model, which
ranges from $3 \times 10^{-3}$ at low $Z^{\prime}$ masses to 1 at $Z^{\prime}$
masses of 8$GeV/c^{2}$.
