The first measurement of the lifetime of the doubly charmed baryon $\Xi_{cc}^{++}$ is presented, with the signal reconstructed in the final state $\Lambda_c^+ K^- \pi^+ \pi^+$. The data sample used corresponds to an integrated luminosity of $1.7\,\mathrm{fb}^{-1}$, collected by the LHCb experiment in proton-proton collisions at a centre-of-mass energy of $13\mathrm{\,Te\kern -0.1em V}$. The $\Xi_{cc}^{++}$ lifetime is measured to be $0.256\,^{+0.024}_{-0.022}{\,\rm (stat)\,} \pm 0.014 {\,\rm(syst)}\mathrm{\,ps}$.