Will the internet die if the power goes out? Not the home router, which one can put on a UPS, but the actual signal?
That would require that the actual ISP loses power, and I think that's very unlikely. More likely is that something else goes wrong and there is a signal outage, but broken cables etc would be more probable there.
During all my time in the DR (since 2010), have I only seen once Claro actually losing all their connectivity for one full day. It was even in the news and the reason was not an outage but a broken cable between Navarrete and POP...
So the practical answer is, no...