Searches for heavy neutral lepton production in \(K^+ \to e^+ N\) and \(K^+ \to \mu^+ N\) decays using the data set collected by the NA62 experiment at CERN in 2016-18 are presented. Upper limits on the elements of the extended neutrino mixing matrix and are established at the levels of and , respectively, improving on the earlier searches for heavy neutral lepton production and decays in the kinematically accessible mass range. Searches for the \(K^+ \to \mu^+ \nu \nu \bar \nu \) and \(K^+ \to \mu^+ \nu X\) decays, where \(X\) is an new light invisible particle, are also reported.