Using pp collision data corresponding to an integrated luminosity of 8.5 fb(-1) recorded by the LHCb experiment at center-of-mass energies of root s = 7, 8, and 13 TeV, the observation of an excited B-c(+) state in the B-c(+)pi(+)pi(-) invariant-mass spectrum is reported. The observed peak has a mass of 6841.2 +/- 0.6(stat) +/- 0.1(syst) +/- 0.8(B-c(+)) MeV/c(2), where the last uncertainty is due to the limited knowledge of the B-c(+) mass. It is consistent with expectations of the B-c*(2(3)S(1))(+) state reconstructed without the low-energy photon from the B-c*(1(3)S(1))(+) -> B-c(+)gamma decay following B-c*(2(3)S(1))(+) -> B-c*(1(3)S(1))(+)pi(+)pi(-). A second state is seen with a global (local) statistical significance of 2.2 sigma (3.2 sigma) and a mass of 6872.1 +/- 1.3(stat) +/- 0.1(syst) +/- 0.8(B-c(+)) MeV/c(2), and is consistent with the B-c(2(1)S(0))(+) state. These mass measurements are the most precise to date.