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K+K− pairs may be produced in photonuclear collisions, either from the decays of photoproduced ϕ(1020) mesons, or directly as non-resonant K+K− pairs. Measurements of K+K− photoproduction probe the couplings between the ϕ(1020) and charged kaons with photons and nuclear targets. We present the first measurement of coherent photoproduction of K+K− pairs on lead ions in ultra-peripheral 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 centre-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.