Where does the driveway go? I do not see a road, just a sump.

Driveway slab ends about there, at that line, and goes to the left. The rest of the driveway is gravel.

You can’t fight Mother Nature; you can only work with her. Obviously there is a low spot right there and the only way to solve this is to provide a lower place for the water to go. Either find a way to route the water to a lower place and/or raise your driveway.

From the picture it looks like your natural water flow is directly back into the picture ie in the direction away from the concrete pad toward the woods.

If so, I believe your water problem can be solved by cutting a swale between the two lines I have put in your picture. It only has to start at the level of your finished driveway, and slope so the water flows away from the driveway down the towards the woods. if you make the swale about 8 to 10' wide and contour it you will be able to mow through it with no problem. You can use the dirt from the swale to as fill for the existing low spot. Note if the standing water is deeper than appear, you may have to make the swale deeper.

When you dig the swale, first did a ditch where the swale will be and get the water flowing properly. The when the weather improves, use the ditch as a gauge for the bottom of the swale.

It is not their land.

In most states in the US the natural water flow can not be interrupted, even if it means that the water drains on to someone else's property who is down hill from you. In this case it appears to be raw undeveloped land. If it is in an addition the developer must develop the properties so that it does not interrupt the natural flow.

Gradually reduce the amount of water as the lawn prepares for winter dormancy.

You need to talk to your county drainage commission (or whatever your local government office is that regulates drainage) and talk to them. I assume you are out in an unincorporated area. If this is a subdivision, you need to talk to the developer as well as the county.

We can't see where or what is connected to your driveway and if there are other lots that will be developed around you. If your driveway is the low spot for the entire area, you are always going to have problems unless drainage ditches or some method of handling that water are installed. If this is an area that is being developed, keep in mind that every new home and paved driveway is going to contribute their rainwater to the problem.

Raise all the concrete please.

You need to see where the water naturally wants to go (that is not towards your house) and then encourage it in that direction. What is to the right of the driveway?