What lies beyond the observable universe?

You've not fully grasped what you have said. You've correctly stated that space itself is expanding but then ask what it is expanding in to. It isn't expanding to anything: it is an expansion _of_ space, not _in_to_ space. When you view the expansion as a property of the space itself (and not anything that may be in that space) the question of what it is expanding into isn't something that arises.