We present upper limits on the production of heavy neutral leptons (HNLs) decaying to $mu pi$ pairs using data collected with the MicroBooNE liquid-argon time projection chamber (TPC) operating at Fermilab. This search is the first of its kind performed in a liquid-argon TPC. We use data collected in 2017 and 2018 corresponding to an exposure of $2.0 times 10^{20}$ protons on target from the Fermilab Booster Neutrino Beam, which produces mainly muon neutrinos with an average energy of $approx 800$ MeV. HNLs with higher mass are expected to have a longer time-of-flight to the liquid-argon TPC than Standard Model neutrinos. The data are therefore recorded with a dedicated trigger configured to detect HNL decays that occur after the neutrino spill reaches the detector. We set upper limits at the $90%$ confidence level on the element $lvert U_{mu4}rvert^2$ of the extended PMNS mixing matrix in the range $lvert U_{mu4}rvert^2<(6.6$-$0.9)times 10^{-7}$ for Dirac HNLs and $lvert U_{mu4}rvert^2<(4.7$-$0.7)times 10^{-7}$ for Majorana HNLs, assuming HNL masses between $260$ and $385$ MeV and $lvert U_{e 4}rvert^2 = lvert U_{tau 4}rvert^2 = 0$.
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