Experimental investigations of tunnelling conductivity oscillations of metal–insulator–gapless semiconductor structures are performed. For the orientation of the magnetic field H | n (n is normal to the structure plane) a splitting of the oscillation maxima of the tunnelling conductivity due to the spin splitting of the Landau levels is observed. It is revealed that when the magnetic field turns away from this direction the amplitudes of the low‐bias maxima decrease monotonously, and at H ⟂ n oscillations corresponding to tunnelling into one spin state only are observed. A theoretical analysis shows that this is caused by an unusual behaviour of the electron wavefunctions near the surface of gapless semiconductors in an applied magnetic field.