Wa'alaykumsalaam
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: With regard to anaesthetic which is applied to a tooth during the day in Ramadaan – do I have to make up that day if I have this anaesthetic?
He replied: No, because anaesthesia does not break the fast. A local anaesthetic only affects the place where it is given, it does not reach the stomach. So if a person is given an anaesthetic when he is observing a naafil or obligatory fast, his fast is still valid.