The recent experimental data obtained by the OBELIX group on $\bar{p}$D and $\bar{p}^4$He total annihilation cross sections are analyzed. The combined analysis of these data with existing antiprotonic atom data allows, for the first time, the imaginary parts of the S-wave scattering lengths for the two nuclei to be extracted. The obtained values are: $Im a^{sc}0 = [- 0.62 \pm 0.02 ({stat}) \pm 0.04 ({sys})] fm$ for $\bar{p}$D and $Im a^{sc}_0 = [- 0.36\pm 0.03({stat})^{+0.19}{-0.11}({sys})] fm$ for $\bar{p}^4$He. This analysis indicates an unexpected behaviour of the imaginary part of the $\bar{p}$-nucleus S-wave scattering length as a function of the atomic weight A: $|Im a^{sc}_0|$ ($\bar{p}$p) > $|Im a^{sc}_0|$ ($\bar{p}$D) > $|Im a^{sc}_0|$ ($\bar{p}^4$He)
