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Within the band gap of a semiconductor no electronic propagating states are allowed, but there exist evanescent states which govern charge transport such as tunneling. In this Letter, we address the issue of their spin dependence in III-V semiconductors. Taking into account the spin-orbit interaction, we treat the problem using a k·p Hamiltonian that we numerically compute for GaAs. Our results show that the removed spin degeneracy in the band gap can lead to giant energy splittings and induces forbidden zones in space where evanescent states are suppressed.