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We use a recent census of the Milky Way (MW) satellite galaxy population to constrain the lifetime of particle dark matter (DM). We consider two-body decaying dark matter (DDM) in which a heavy DM particle decays with lifetime $τ$ comparable to the age of the Universe to a lighter DM particle (with mass splitting $ε$) and to a dark radiation species. These decays impart a characteristic "kick velocity," $V_{\mathrm{kick}}=εc$, on the DM daughter particles, significantly depleting the DM content of low-mass subhalos and making them more susceptible to tidal disruption. We fit the suppression of the present-day DDM subhalo mass function (SHMF) as a function of $τ$ and $V_{\mathrm{kick}}$ using a suite of high-resolution zoom-in simulations of MW-mass halos, and we validate this model on new DDM simulations of systems specifically chosen to resemble the MW. We implement our DDM SHMF predictions in a forward model that incorporates inhomogeneities in the spatial distribution and detectability of MW satellites and uncertainties in the mapping between galaxies and DM halos, the properties of the MW system, and the disruption of subhalos by the MW disk using an empirical model for the galaxy--halo connection. By comparing to the observed MW satellite population, we conservatively exclude DDM models with $τ< 18\ \mathrm{Gyr}$ ($29\ \mathrm{Gyr}$) for $V_{\mathrm{kick}}=20\ \mathrm{km}\, \mathrm{s}^{-1}$ ($40\ \mathrm{km}\, \mathrm{s}^{-1}$) at $95\%$ confidence. These constraints are among the most stringent and robust small-scale structure limits on the DM particle lifetime and strongly disfavor DDM models that have been proposed to alleviate the Hubble and $S_8$ tensions.