Exclusive ${{{\uprho _{}^{}} _{}^{}}{{\left( {770}\right) }{}_{}^{}}} ^{0}$ photoproduction is measured for the first time in ultraperipheral pPb collisions at $\sqrt{\smash [b]{s_{_{\mathrm {NN}}}}} = 5.02\,\text {Te}\text {V} $ with the CMS detector. The cross section $\sigma ({\upgamma _{}^{}} \mathrm{p}\rightarrow {{{\uprho _{}^{}} _{}^{}}{{\left( {770}\right) }{}_{}^{}}} ^{0}\mathrm{p})$ is $11.0 \pm 1.4\,\text {(stat)} \pm 1.0\,\text {(syst)} $ $\mu $ b at $\langle W_{{\upgamma _{}^{}} \mathrm{p}}\rangle = 92.6\,\text {Ge}\text {V} $ for photon–proton centre-of-mass energies $W_{{\upgamma _{}^{}} \mathrm{p}}$ between 29 and $213\,\text {Ge}\text {V} $ . The differential cross section $\mathrm {d}\sigma /\mathrm {d}|t |$ is measured in the interval $0.025< |t | < 1\,\text {Ge}\text {V} ^{2}$ as a function of $W_{{\upgamma _{}^{}} \mathrm{p}}$ , 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 $\sigma ({\upgamma _{}^{}} \mathrm{p}\rightarrow {{{\uprho _{}^{}} _{}^{}}{{\left( {770}\right) }{}_{}^{}}} ^{0}\mathrm{p})$ has a power-law dependence on the photon–proton centre-of-mass, consistent with electron–proton collision measurements performed at HERA. The $W_{{\upgamma _{}^{}} \mathrm{p}}$ dependence of the exponential slope of the differential cross section $\mathrm {d}\sigma /\mathrm {d}|t |$ is also measured.