A search for the lepton number violating $K^{+}\rightarrow\pi^{-}\mu^{+}e^{+}$ and lepton flavour violating $K^{+}\rightarrow\pi^{+}\mu^{-}e^{+}$ decays is reported using NA62 data collected in 2017 and 2018. No significant excess is observed above the background expectation and therefore upper limits are established on the branching ratios at $90\%$ confidence level: $\mathcal{B}(K^{+}\rightarrow\pi^{-}\mu^{+}e^{+})<4.2\times10^{-11}$ and $\mathcal{B}(K^{+}\rightarrow\pi^{+}\mu^{-}e^{+})<6.6\times10^{-11}$. These results improve over the previous limits by factors of $12$ and $8$ respectively.