The answer to this is always persevere, but do it smartly.
Every business will sometimes experience difficulties in addressing a customer’s dissatisfaction or complaint. There’s sometimes a big temptation to just give up and focus on finding new customers. Be warned this may lead to bigger problems, it’s important therefore to resist this temptation.
Unhappy customers are likely to tell their friends about their experience and at times utilise social media to air their grievance. Even worse, they might get a newspaper or television outlet involved. This will make your efforts to find those new customers much more difficult.
So even if the customer is being completely unreasonable, don’t argue. Instead, make every effort to apologise, and most importantly find out how to put things right and make amends. It is better to resolve the customer's issue and conclude on a positive note then to risk the damage an unhappy customer can do to even the best planned marketing stratefy.