Yes you can, providing you have a Personal Public Service Number (PPSN). A PPSN is an explicit requirement for investing in State Savings including Prize Bonds.
You must provide proof of your PPSN and also proof of name, address and date of birth upon request.
If you do not have a PPSN and wish to see if you are eligible to get one, you can do so by applying to the Department of Employment Affairs and Social Protection. State Savings cannot advise on the requirements and/or process in regard to applications for PPSN.
Foreign Tax Identification Numbers are not accepted by State Savings.