What is the ruling on shunning a neighbor due to his backbiting, knowing that he is ill-mannered and openly commits sins, and does the Prophet's (PBUH) saying about deeds not being raised apply to shunning?

It is permissible to shun an annoying and wicked neighbor if that deters them or repels their harm. Such shunning is not considered broken if one returns their greeting of peace (salam), and this is the view of the majority of scholars. The prohibition against shunning for more than three nights applies only to those who shun for personal or worldly reasons. As for shunning people of innovation (bid'ah) and wickedness (fusuq), it is always permissible. It is not obligatory to return the greeting of peace to a wicked person, nor is it recommended to initiate it, if refraining from doing so serves as a reprimand for them.