Is it necessary for one who broke a wooden comb while combing another person’s hair with it to pay its value, especially since the one who broke it is ashamed to ask for its value back?

If you used the comb without its owner's permission, you must return a similar one to her, unless she absolves you or you reach a settlement. However, if you used it with her permission and did not exceed the agreed terms, you are not liable. Your promise to give her a comb in its place is a non-binding promise, so there is no sin upon you for not fulfilling it. However, it is preferable to fulfill it, especially if she desires the replacement.