We searched for the bound state of the neutron-rich -hypernucleus ⁶_ΛH, using the ⁶Li(π⁻ ,K ⁺)X double charge-exchange reaction at a π − beam momentum of 1.2 GeV/c at J-PARC. A total of 1.4 × 10¹²_π− was driven onto a ⁶Li target of 3.5-g/cm² thickness. No event was observed below the bound threshold, i.e., the mass of ⁴_ΛH + 2n, in the missing-mass spectrum of the ⁶Li(π − ,^K+)X reaction in the 2° < θ_πK < 20° angular range. Furthermore, no event was found up to 2.8 MeV/c² above the bound threshold. We obtained the doubledifferential cross section spectra of the ⁶Li(π⁻ ,K⁺)X reaction in the angular range of 2°< θπK < 14°. An upper limit of 0.56 nb/sr (90% C.L.) was obtained for the production cross section of the ⁶_ΛH hypernucleus bound state. In addition, not only the bound state region, but also the  Λ continuum region and part of the ∑⁻ quasifree production region of the ⁶Li(π⁻ ,K⁺) reaction were obtained with high statistics. The present missing-mass spectrum will facilitate the investigation of the  ∑^-nucleus optical potential for  ∑^--⁵He through spectrum shape analysis.