Is it permissible to give Zamzam water to a Christian neighbor with the intention of healing?

There is no harm in gifting Zamzam water to a non-believer, especially if it serves a predominant interest, such as inclining them towards Islam. There is no reported prohibition preventing a non-believer from drinking from Zamzam. Indeed, the Prophet, peace and blessings be upon him, informed us that it contains healing from ailments, saying: "Zamzam is a delicious food and a healing from sickness," and this applies to both Muslims and non-believers.