As of March 2015, it is only possible to obtain a free PSP as a gift from a friend or family member. There is a very small chance that one may be offered through a "freecycling" community, however.
Freecycling communities usually connect through email groups or message boards to offer each other all sorts of unwanted items. It's possible that a PSP may be offered through one of these, though not very likely as they still have significant resale value.
Sony stopped manufacturing new PSPs as of 2014, so it is extremely unlikely that they will give any away in contests or promotions. Used PSPs can still be purchased through sites such as Amazon and eBay for well under 100 dollars.
While it is unlikely that a free PSP system can be found, it is possible to get a number of free "homebrew" games for the system. These games are made by individual developers and are legally distributed for free, though they usually require the use of custom firmware. Some examples of free games of this nature are "Touhou", "Nazi Zombies Portable", "Naruto Ultimate Ninja Battle", "Stickman Jump" and "Cube Runner Advanced." There is also a free "Minecraft" clone for PSP called 'Lamecraft."