When you lose a customer, is it possible to win them back?

Yes, it is possible, but don't even try to win a customer back unless you are ready to have them back. 

They left you for a reason. Fix the issue that caused them to leave in the first place. Only then, should you try to win a customer back. 

And when you know you are ready to have them back, treat them like a VIP (very individual person). If you are in a position to do so, go see him or her in person. Don't settle for sending an email or text message. Definitely, don't wait for them to come to see you. It won't happen. And it is your responsibility to fix it. Own your mistake, face to face.  

Once I was an hour late for a speech. Immediately I told my customer I would deduct $5,000 from the speaking fee. They didn't ask for the fee reduction. They didn't expect it. But it was the right thing to do. I didn't serve the customer to the level I should have. It needed to be handled professionally. 

Professionals want to take a bad situation and make it right. 

So if you want to win a customer back, you need to fix the problem first. Then be professional and personal. You might win the customer back.