My recommendation because of your description is to hire a professional landscaper to help resolve this problem.

When it comes to drainage problems, one not only needs to know about the area of the problem. They must have an overview understanding of the grade of the entire yard (any portion that may help in contributing to the problem) and an understanding of the grade SURROUNDING the yard -- which is usually other people's property. Why ...?? Because knowing only about the problem itself isn't enough information to know about the options of where excess water can be directed in order to solve the problem. The drainage options of the whole surrounding area must be known so that we know where and how to put the water such that it can't possibly come back or do damage elsewhere.

That said, it sounds like a "sticky" situation with limited options. In general, if there is a lower not too far away, a drain/catch basin, and grading to it, can be installed and the water gravity piped to the nearby lower area. If there is not a nearby area, the water can still be collected the same way, but it might need to be pumped to a different location with a mechanical pump. Either option could be simple or complicated depending on the surrounds (which we don't know.)