A search for heavy neutral lepton production in $K^{+}$ decays using a data sample collected with a minimum bias trigger by the NA62 experiment at CERN in 2015 is reported. Upper limits at the ${10}^{-7}$ to ${10}^{-6}$ level are established on the elements of the extended neutrino mixing matrix $|U_{e4}{|}^2$ and $|U_{\mu 4}{|}^2$ for heavy neutral lepton mass in the ranges 170–448 MeV/$c^2$ and 250–373 MeV/$c^2$, respectively. This improves on the previous limits from HNL production searches over the whole mass range considered for $|U_{e4}{|}^2$, and above 300 MeV/$c^2$ for $|U_{\mu 4}{|}^2$.