Yes. For example, you can get a work permit once as a citizen of the United Kingdom and a second time as an Irish citizen
To do this, you need to:
- have a valid passport issued by the country you're applying under,
- meet all of the eligibility criteria for this country, and
- provide a letter of explanation in your work permit application that says
- you are a dual citizen
- your other work permit is under your other country of citizenship.