Don't know about free but if you look on Preloved etc you'll probably find loads of slightly older puppies that aren't too expensive (especially if mixed breed). Sadly I think a lot of people get them at 8 weeks but by the time they are a few months old and the people are sick of 'accidents' on the carpet they get rid of them.
Be prepared for veterinary costs though if you go down this route, someone selling on a pup cheap because they don't want it any more are unlikely to have taken its health into much consideration - you may have to fork out for vaccinations, wormers, flea treatments etc.
We got ours this way at 4 months old, her family didn't want her any longer so were selling her on. She was cheap but hadn't been vaccinated and was terribly full of worms so it was a few vet visits to put her right.
Worth it though, she's nearly 18 months now and she's great, we love her very much
Alternatively you could try a rescue centre, it'd probably cost more at the outset for the adoption fee but you'd be more sure of having a healthy pup that's already had its vaccinations etc which would probably be less stressful.