K^{+}K^{-} pairs may be produced in photonuclear collisions, either from the decays of photoproduced ϕ(1020) mesons or directly as nonresonant K^{+}K^{-} pairs. Measurements of K^{+}K^{-} photoproduction probe the couplings between the ϕ(1020) and charged kaons with photons and nuclear targets. The kaon-proton scattering occurs at energies far above those available elsewhere. We present the first measurement of coherent photoproduction of K^{+}K^{-} pairs on lead ions in ultraperipheral collisions using the ALICE detector, including the first investigation of direct K^{+}K^{-} production. There is significant K^{+}K^{-} production at low transverse momentum, consistent with coherent photoproduction on lead targets. In the mass range 1.1<M_{KK}<1.4 GeV/c^{2} above the ϕ(1020) resonance, for rapidity |y_{KK}|<0.8 and p_{T,KK}<0.1 GeV/c, the measured coherent photoproduction cross section is dσ/dy=3.37±0.61(stat)±0.15(syst) mb. The center-of-mass energy per nucleon of the photon-nucleus (Pb) system W_{γPb,n} ranges from 33 to 188 GeV, far higher than previous measurements on heavy-nucleus targets. The cross section is larger than expected for ϕ(1020) photoproduction alone. The mass spectrum is fit to a cocktail consisting of ϕ(1020) decays, direct K^{+}K^{-} photoproduction, and interference between the two. The confidence regions for the amplitude and relative phase angle for direct K^{+}K^{-} photoproduction are presented.