We searched for the bound state of the neutron-rich $Lambda$-hypernucleus ${}^{6}_{Lambda}$H, using the ${}^{6}$Li($pi^{-}, K^{+}$)X double charge-exchange reaction at a $pi^{-}$ beam momentum of 1.2 GeV/c at J-PARC. A total of $1.4 times 10^{12}$ $pi^{-}$ was driven onto a ${}^{6}$Li target of 3.5-g/$rm cm^2$ thickness. No event was observed below the bound threshold, i.e., the mass of ${}^{4}_{Lambda}$H + 2n, in the missing-mass spectrum of the ${}^{6}$Li($pi^{-}, K^{+}$)X reaction in the $2^{circ}$ < $theta_{pi K}$ < $20^{circ}$ angular range. Furthermore, no event was found up to 2.8 MeV/$c^2$ above the bound threshold. We obtained the the double-differential cross section spectra of the ${}^{6}$Li($pi^{-}, K^{+}$)X reaction in the angular range of $2^{circ}$ < $theta_{pi K}$ < $14^{circ}$. An upper limit of 0.56 nb/sr (90% C.L.) was obtained for the production cross section of the ${}^{6}_{Lambda}$H hypernucleus bound state. In addition, not only the bound state region, but also the $Lambda$ continuum region and part of the $Sigma^{-}$ quasi-free production region of the ${}^{6}$Li($pi^{-}, K^{+}$)X reaction, were obtained with high statistics. The present missing-mass spectrum will facilitate the investigation of the $Sigma^{-}$-nucleus optical potential for $Sigma^{-}$-${}^{5}$He through spectrum shape analysis.
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