Yes, international students have the option to extend their stay in Ireland after completing their postgraduate studies through various visa and immigration routes. Here are some options available:

1. Graduate Route Permission: International students who have completed a recognized degree at Level 9 or above on the National Framework of Qualifications (NFQ) can apply for the Graduate Route Permission. This allows them to stay in Ireland for up to 24 months to seek employment or start a business. To be eligible, students must have a valid Stamp 2 student permission at the time of application and meet certain other criteria.
2. Critical Skills Employment Permit: International graduates who secure employment in Ireland in a role that falls under the Critical Skills Occupations List may be eligible to apply for a Critical Skills Employment Permit. This permit allows them to work in Ireland for an initial period of up to two years, with the possibility of renewing the permit thereafter. It also provides a pathway to long-term residency and citizenship.
3. General Employment Permit: If the graduate’s job offer does not qualify for the Critical Skills Employment Permit but is in a skilled occupation, they may still be eligible to apply for a General Employment Permit. This permit allows them to work in Ireland for up to two years, with the possibility of renewal.
4. Stamp 1G Permission: International graduates who secure graduate employment in Ireland may be eligible for Stamp 1G permission, which allows them to work in the country for the duration of their employment contract, up to a maximum of two years. This permission is typically granted to graduates who secure employment before their student permission expires.
5. Further Study: Some international students may choose to pursue further studies in Ireland by enrolling in another academic program, such as a doctoral (Ph.D.) program or a postdoctoral research position. In such cases, they can apply for a student visa or permission to stay in Ireland for the duration of their new program of study.

It’s important for international students to familiarize themselves with the specific visa and immigration requirements and seek advice from the Irish Naturalisation and Immigration Service (INIS) or an immigration lawyer to determine the best option for extending their stay in Ireland after completing their postgraduate studies.