We have carried out an experiment to search for an eutron-rich hypernucleus, Λ6H, by the 6Li(π−,K+) reaction at pπ−=1.2 GeV/c. The obtained missing-mass spectrum with an estimated energy resolution of 3.2MeV (FWHM) showed no peak structure corresponding to the Λ6H hypernucleus neither below nor above the Λ4H+2n particle decay threshold. An upper limit of the production cross section for the bound Λ6H hypernucleus was estimated to be 1.2nb/sr at 90% confidence level.