Is the defamation I suffered due to an accusation against my honor a punishment from Allah? Is it Islamically permissible for me to seek revenge against those responsible, in kind or by killing, in defense of my honor? And what is the solution to this problem?

If you are innocent of the accusations made against you, then praise Allah for that, and be patient and seek your reward from Him. In the lives of the righteous predecessors, you have an excellent example, for whoever fears Allah and is patient, He will make a way out for him and record a reward for him. However, if you have committed what was attributed to you, then hasten to sincere repentance. And upon the one who accused you is concealment and counsel. Do not be concerned with what has passed, and occupy yourself with what benefits you. It is not permissible for you to take revenge yourself. If you wish to claim your right, then resort to the competent authorities.