What can I do about the wall next to my dishwasher?

You may want to consider just repainting with a higher sheen paint. The glossier the paint, the more resistant it is to stains and the easier it is to wipe down. If it's a matte finish, consider switching to a semi-gloss or a satin. They also make matte paints that are designed to be scrubbed down on a regular basis. Most paint companies have a "scrubable" line up.

I would paint it with paint that is easy to wash, a semi gloss or gloss paint. Or attach a decorative panel that is a material that is easy to clean. I have this same issue around my trash can. My kids appear to think that the general direction of the trash can works and the wall around it gets quite messy. I did a bit of bead board that I did a high gloss finish, and I made sure it was thick enough to cover the seams so things didn't get stuck in there, so I got the look of bead board without having to worry about scrubbing it.

Wipe it down after loading the dishwasher. Who is doing the dishes? How much splish splashing is going on? I do dishes several times a day and my walls are not covered in food. Maybe the dishwashing person needs to slow down, pay attention to what they are doing to make less mess? After doing the dishes, I wipe all the counters down, clean the stove and wipe the cabinets down that are located next to the stove. Takes only a few moments to keep things neat, tidy and clean.

Tile or a washable panel like for a shower

TILE wipes off great.