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<𝑀𝐾𝐾<1.4  GeV/𝑐2 above the 𝜙⁡(1020) resonance, for rapidity |𝑦𝐾𝐾|<0.8 and 𝑝𝑇,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 𝑊𝛾Pb,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.