  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 $|U_{\ell
4}|^2$ ($\ell=e,\mu$) for heavy neutral lepton mass in the range $170-448~{\rm
MeV}/c^2$. This improves on the results from previous production searches in
$K^+$ decays, setting more stringent limits and extending the mass range.
