We have measured, for the first time, the inclusive missing-mass spectrum of the C-12(K-,p) reaction at an incident kaon momentum of 1.8 GeV/c at the J-PARC K1.8 beamline. We observed a prominent quasi-elastic peak (K- p -> K- p) in this spectrum. In the quasi-elastic peak region, the effect of secondary interaction is apparently observed as a peak shift, and the peak exhibits a tail in the bound region. We compared the spectrum with a theoretical calculation based on the Green's function method by assuming different values of the parameters for the (K) over bar -nucleus optical potential. We found that the spectrum shape in the binding-energy region -300 MeV < B-K < 40 MeV is best reproduced with the potential depths V-0 = -80 MeV (real part) and W-0 = -40 MeV (imaginary part). On the other hand, we observed a significant event excess in the deeply bound region around B-K similar to 100 MeV, where the major decay channel of K- NN -> pi Sigma N is energetically closed, and the non-mesonic decay modes (K- NN -> Lambda N and Sigma N) should mainly contribute. The enhancement is fitted well by a Breit-Wigner function with a kaon-binding energy of 90 MeV and width 100 MeV. A possible interpretation is a deeply bound state of a Y*-nucleus system.