[en] Nonsinglet contributions to the g₁(x, Q²) structure function are calculated in the double-logarithmic approximation of perturbative QCD in the region x<<1. Double logarithmic contributions of the type (α_sln²(1/x))^k which are not included in the GLAP evolution equations are shown to give a stronger rise at small-x than the extrapolation of the GLAP expressions. Further enhancement in the small-x region is due to non-ladder Feynman graphs which in the DLA of the unpolarized structure functions do not contribute. Compared to the conventional GLAP method (where neither the whole kinematical region which gives the double logs nor the non-ladder graphs are taken into account) our results lead to a growth at small-x which, for HERA parameters, can be larger by up to factor of 10 or more. (orig.)