High-spin states in $^{71}\mathrm{As}$ were studied using the $^{54}\mathrm{Fe}$$({}^{23}$Na,$\ensuremath{\alpha}2p)$ reaction at 80 MeV. Prompt $\ensuremath{\gamma}$-$\ensuremath{\gamma}$ coincidences were measured using the Florida State University Compton-suppressed Ge array consisting of three clover detectors and seven single-crystal detectors. The existing high-spin level scheme has been verified, and 21 new transitions have been added based on an investigation of weak $\ensuremath{\gamma}$-ray coincidence relations and relative $\ensuremath{\gamma}$-ray intensities. Lifetimes of 16 excited states were measured using the Doppler-shift attenuation method applied to the experimental line shapes of decays in all of the known rotational bands. The $B(E2)$ strengths inferred from the lifetimes indicate that moderate to high collective behavior persists to the highest observed spins in the lowest positive- and negative-parity bands, in qualitative agreement with projected shell-model calculations. The band suggested to be based on the $\ensuremath{\pi}{f}_{7/2}$ orbital shows a similar degree of collectivity within the same spin range, with $B(E2)$ values in good agreement with those predicted by the projected shell model assuming a constant prolate deformation of ${\ensuremath{\epsilon}}_{2}=+0.27$. The experimental ${Q}_{t}$ values in this band are somewhat smaller than predicted by cranked Woods-Saxon calculations.