The NA48/2 experiment at CERN collected a large sample of charged kaon decays to final states with multiple charged particles in 2003–2004. A new upper limit on the rate of the lepton number violating decay K±→π∓μ±μ± is reported: B(K±→π∓μ±μ±)<8.6×10−11 at 90% CL. Searches for two-body resonances X in K±→πμμ decays (such as heavy neutral leptons N4 and inflatons χ) are also presented. In the absence of signals, upper limits are set on the products of branching fractions B(K±→μ±N4)B(N4→πμ) and B(K±→π±X)B(X→μ+μ−) for ranges of assumed resonance masses and lifetimes. The limits are in the (10−11,10−9) range for resonance lifetimes below 100 ps.