No. The Sims Castaway Stories on PSP does not include pregnancy, babies, or child-rearing. The game strips the life simulation down to survival, exploration, and relationship-building among the adult castaways.
The PSP hardware (333 MHz MIPS, 32 MB RAM) couldn't support the full Sims 2 simulation engine. The "Stories" sub-series (Castaway, Life, Pets) were built on a stripped-down engine optimized for directed gameplay and lower specs. All console versions (PS2, Wii, PSP) share this limitation.
Castaway Stories is a survival romance adventure—not a generational legacy builder. Enjoy it for what it is: a focused, portable Sims experience with an actual ending.