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