The NA48/2 experiment at CERN collected a large sample of charged kaon decays into final states with multiple charged particles in 2003--2004. A new upper limit on the rate of the lepton number violating decay $K^{\pm}\to\pi^{\mp}\mu^{\pm}\mu^{\pm}$ obtained from this sample is reported: $\mathcal{B}(K^{\pm}\to\pi^{\mp}\mu^{\pm}\mu^{\pm})<8.6 \times 10^{-11}$ at 90\% CL. Searches for two-body resonances in the $K^{\pm}\to\pi\mu\mu$ decays (including heavy neutral leptons~$N_4$ and inflatons~$\chi$) in the accessible range of masses and lifetimes are also presented.
In the absence of a signal, upper limits are set on the products of branching ratios~$\mathcal{B}(K^{\pm}\to\mu^{\pm}N_4)\mathcal{B}(N_4\to \pi \mu)$ and $\mathcal{B}(K^{\pm}\to\pi^{\pm}\chi)\mathcal{B}(\chi\to\mu^+\mu^-)$ as functions of the resonance mass and lifetime. These limits are in the $10^{-10}-10^{-9}$ range for resonance lifetimes below 100~ps.