Is it permissible to continue corresponding with a non-Muslim woman and giving her gifts after knowing that she is Buddhist?

It is permissible to correspond and exchange gifts with a non-Muslim woman, as long as she has not fought Muslims or expelled them from their homes. This is based on the Almighty's saying: "Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them." It is desirable to use this relationship to invite her to Islam.