According to the theory of relativity, time doesn’t slow down because you’re going further out into space, it slows down as you increase your speed. The closer you get to the speed of light, the slower your clock will run (to observers outside your frame of reference). For the person increasing their speed, however, time will appear to be functioning as normal. Likewise, your mass will also increase, and your dimensions will become elongated and stretched out. This is why one can never hit the speed of light, because at that speed, your mass would become infinite, and there isn’t enough energy in the universe to accelerate something with an infinite mass. (Note on shows like “Star Trek” that when they go to warp speed, they portray the Enterprise as actually stretching for a moment as it accelerates to speed.)
So, in answer to your question, the universe is simply expanding into nothing. However, you are correct on one point. Time is integrally related to space. In other words, they are part of the same thing. So, beyond our universe, there is no time. As our universe expands, it brings time along with it, because you can’t have one without the other.