Need ideas about any plants that will grow in a swale that is very mud

instead of fighting with your issue make assorted container plantings based on your zone

Do you know what planting zone you're located in?

Here's how to figure it out: https://garden.org/nga/zipzone/

I planted a couple of willow trees in an area of my yard that was excessively wet. They are thirsty plants and really helped to dry it up.

All of the previous answers are good. Many of the plants that are used in and around the ponds or fish ponds people have in their yards are excellent in muddy areas. One example is the Iris that come in beautiful colours of purple, rust, yellow, white, black and other colours. Creeping Jenny which spreads, has delicate green leaves and lovely little yellow flowers. Tree of Heaven (ILantis) and is a tree spreads like its relative , the the Sumac bush which grows wild and spreads. These grow in very dry or very wet areas. Hope your garden are turns out well!! Poppies grow in almost any soil wet or dry.

I would say cat tails. I can't think of any others. J.