I would use a white cloth and just a bit of dawn dishsoap in a pail of water.

If you don't want to see marks when dry you do probably need to clean the entire wall.

One traditional cleaner for walls in non-sudsing ammonia. It must be rinsed though which means two steps. Often people recommend TSP. It also needs to be rinsed I think.

I use Dirtex. It was recommended to me years ago by a professional painter. It works very well and doesn't need to be rinsed like other products. The other advantage is that doesn't have a strong, if any, odor. It comes in a powder. You can order it at Amazon. It can also be used for other cleaning needs.