No.
The island in question has two countries - Ireland and Northern Ireland.
Northern Ireland is part of the UK.
Ireland is a totally different, entirely independent country that is also part of the EU.
Due to the history of Ireland/NI, the long conflict and the border between the two we have the Good Friday Agreement. The GFA effectively brought peace to Northern Ireland, and part of the GFA was that there should be no hard border between North and South.
If the UK leaves the EU as planned, along with leaving the Single Market and removing the right of freedom of movement it would in theory require a hard border between Ireland and Northern Ireland (UK), as Northern Ireland would not be in the EU and therefore new border customs, immigration etc rules would apply, meaning a hard border.
That would break the GFA and put the Peace Process in jeopardy.
EDITED: 17 Dec 2017 15:59 by ANT_THOMAS