I'm settling a case outside the court and without involving lawyers, is it legally permissible to ask for punitive damages in the friendly agreement contract?
Wait, this is redundant; you want punitive damages in a "friendly agreement." Punitive damages are to punish the person for what they did, hence the name punitive; there is nothing friendly about it. No rational person settling a possible case is going to agree to that. You may get your actual damages but if you want punitive damages get a lawyer and sue in Court.
You can legally ask for anything you want, but there is nothing friendly about punitive damages.