Exclusive ρ(770)0 photoproduction is measured for the first time in ultraperipheral pPb collisions at sNN=5.02Te with the CMS detector. The cross section σ(γp→ρ(770)0p) is 11.0±1.4(stat)±1.0(syst)μb at ⟨Wγp⟩=92.6Ge for photon–proton centre-of-mass energies Wγp between 29 and 213Ge. The differential cross section d σ/ d | t| is measured in the interval 0.025<|t|<1Ge2 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 d σ/ d | t| is also measured.