  The $S=1/2$ antiferromagnetic Heisenberg spin chain compound Sr$_2$CuO$_3$
doped with $1 \%$ and $2 \%$ of Ni impurities has been studied by means of
$^{63}$Cu nuclear magnetic resonance. A strong decrease of the spin-lattice
relaxation rate $T_1^{-1}$ at low temperatures points toward a spin gap, while
a stretching exponent $\lambda < 1$ and a frequency dependence of $T_1^{-1}$
indicate that this spin gap varies spatially and should rather be characterized
as a spin pseudogap. The magnitude of the spin pseudogap scales with doping
level. Our results therefore evidence the finite-size character of this
phenomenon. Moreover, an unusual narrowing of the low temperature NMR lines
reveals the suppression of the impurity-induced staggered paramagnetic response
with increasing doping level.
