I am not aware of a place in the Bible where violence is expressly forbidden. I'm aware of commandments to not murder and to leave vengeance to God, but self-defense doesn't meet the definition of vengeance. Vengeance is defined as: "infliction of injury, harm, humiliation, or the like, on a person by another who has been harmed by that person; violent revenge"
So vengeance is revenge. Stopping someone from hurting someone isn't vengeance. It's called self-defense, a preventative measure to keep us safe. I would suggest that self-defense is Biblical.
The proper use of self-defense has to do with wisdom, understanding, and tact. In Luke 22:36, Jesus tells His remaining disciples, “If you don’t have a sword, sell your cloak and buy one.” Jesus knew that now was the time when His followers would be threatened, and He upheld their right to self-defense. Even though Peter cut off someone's ear in this chapter and was rebuked by Jesus for it, he was standing in the way of God's will trying to protect Jesus from being arrested. Jesus had already told them numerous times at this point that he will be arrested and crucified for the sins of the world.
In summary, I believe people are allowed to defend themselves but shouldn't do it in every situation. James 3:17 says we can pray for wisdom from heaven to ask God to give us judgement on when to act or not. If at all possible, live in peace with all people.