Yes, the banks will support you. We specialise in this area. The banks that we work with, support your income if you are on Stamp 1, Stamp 1G, Stamp 4 and Stamp 5.
There can be some deposit restrictions with some banks depending on how long you are in Ireland, however it is possible to obtain a mortgage in Ireland under the normal Irish lending rules.
You are also allowed to obtain a gift from overseas, although the banks will always assess any gifts depending on the amount of the gift, from what country the gift is from and how the giftor is related to you.