A search for heavy neutral lepton (N) production in $K^{+}\to e^{+}N$ decays using the data sample collected by the NA62 experiment at CERN in 2017–2018 is reported. Upper limits of the extended neutrino mixing matrix element $|U_{e4}{|}^2$ are established at the level of ${10}^{-9}$ over most of the accessible heavy neutral lepton mass range 144–462 MeV/$c^2$, with the assumption that the lifetime exceeds 50 ns. These limits improve significantly upon those of previous production and decay searches. The $|U_{e4}{|}^2$ range favoured by Big Bang Nucleosynthesis is excluded up to a mass of about 340 MeV/$c^2$.