The bond-disordered Kitaev model attracts much attention due to the experimental relevance in α-RuCl3 and A3LiIr2O6 (A = H, D, Ag, etc.). Applying a magnetic field to break the time-reversal symmetry leads to a strong modulation in mass terms for Dirac cones. Because of the smallness of the flux gap of the Kitaev model, a small bond disorder can have large influence on itinerant Majorana fermions. The quantization of the thermal Hall conductivity κxy/T disappears by a quantum Hall transition induced by a small disorder, and κxy/T shows a rapid crossover into a state with a negligible Hall current. We call this immobile liquid state Anderson–Kitaev spin liquid (AKSL). Especially, the critical disorder strength δJc1 ~ 0.05 in the unit of the Kitaev interaction would have many implications for the stability of Kitaev spin liquids.