We have measured an inclusive missing-mass spectrum of the 12 C(K-, p) reaction over wide energy region at the kaon incident momentum of 1.8 GeV/c at J-PARC K1.8 beam line. This measurement was carried out as a by-product of a pilot run of J-PARC E05 experiment, which searched for the Ξ-hypernuclei via 12C(K-, K+) reaction. We have compared the observed missing-mass spectrum with a DWIA calculation to study the K̄-nucleus interaction. In this comparison, we have changed the depth of both real and imaginary parts of the optical potential between the K̄ and nucleus. It is found that the observed spectrum is best reproduced by the calculation of the potential depth of V0 ∼-80 MeV (real part) and W0 ∼-40 MeV (imaginary part). On the other hand, we have observed a definitive excess around the deeply bound region BK ≳ 100 MeV. The excess cannot be explained by the any optical potential and quasi-free background reaction. It is well fitted with a Breit-Wigner function with a binding energy of 90 MeV and a width of 100 MeV, which can be interpreted as a deeply bound Λ(1405) nucleus.