The exact solution at zero boundary condition of the problem of the carrier in a spatially confined gapless semiconductor in a magnetic field parallel to the surface is presented. It is shown that, because the energy spectrum of a gapless semiconductor is formed by a strong relativistic spin-orbit interaction, the space confinement leads to the effective attraction of the electron to the surface, which is different for different spin states. As a result, the energy of Landau levels in the case of one-side confinement of a semiconductor can even diminish when the oscillator centre is shifted towards the surface. The manifestation of the considered effect in tunnelling experiments is discussed. By solving the same problem in a gapless semiconductor film it is shown that the most striking effect, which has the same origin, is the confluence of the lowest electronic levels a₁ and b₁ and inversion of the sign of the g-factor for states with n>1 as the ratio of film width to the magnetic length is reduced. The possibility of the experimental testing of this phenomenon is discussed.