Using proton-proton collision data, collected with the LHCb detector and corresponding to 1.0, 2.0 and 1.9 fb(-1) of integrated luminosity at the centre-of-mass energies of 7, 8, and 13 TeV, respectively, the decay Lambda(0)(b) -> chi(c1)(3872)pK(-) with chi(c1)(3872) -> J/psi pi(+)pi(-) is observed for the first time. The significance of the observed signal is in excess of seven standard deviations. It is found that (58 +/- 15)% of the decays proceed via the two-body intermediate state chi(c1)(3872)Lambda(1520). The branching fraction with respect to that of the Lambda(0)(b) -> psi(2S)pK(-) decay mode, where the psi(2S) meson is reconstructed in the J/psi pi(+)pi(-) final state, is measured to be:
B(Lambda(0)(b) -> chi(c1)(3872)pK(-))/B (Lambda(0)(b) -> psi(2S)pK(-)) x B(chi(c1)(3872) -> J/psi pi(+)pi(-))/B(psi(2S) -> J/psi pi(+)pi(-)) = (5.4 +/- 1.1 +/- 0.2) x 10(-2),
where the first uncertainty is statistical and the second is systematic.