Exclusive ρ(770)0 photoproduction is measured for the first time in ultraperipheral pPb collisions at 𝑠NN‾‾‾√=5.02TeV with the CMS detector. The cross section 𝜎(γp→ρ(770)0p) is 11.0±1.4(stat)±1.0(syst) 𝜇 b at ⟨𝑊γp⟩=92.6GeV for photon–proton centre-of-mass energies 𝑊γp between 29 and 213GeV . The differential cross section d𝜎/d|𝑡| is measured in the interval 0.025<|𝑡|<1GeV2 as a function of 𝑊γp , where t is the squared four-momentum transfer at the proton vertex. The results are compared with previous measurements and theoretical predictions. The measured cross section 𝜎(γp→ρ(770)0p) has a power-law dependence on the photon–proton centre-of-mass, consistent with electron–proton collision measurements performed at HERA. The 𝑊γp dependence of the exponential slope of the differential cross section d𝜎/d|𝑡| is also measured.