How to remove water based wood stain splashed onto painted surface
The primary difficulty you are facing is that virtually all paint is somewhat porous, water based paints generally more so than alkyd (so-called oil) based paints. It is complicated by the fact that the stain that has been added is also water based. While there are plasiticizers in water based paints used to make them more stain resistant, these only partially prevent absorption of staining materials.

Soap and water is a good first start for most stains (after gentle blotting). The next step is usually a stronger detergent based cleanser, such as Fantastic. A test should be made on an inconspicuous area before trying any cleanser. If the test does not indicate serious compromise of the paint, proceed to attack the stain.

First put a bit of the cleanser on a paper towel or clean rag, and try rubbing the spot. If that does not work (or work well enough), try spraying directly on the stain and rubbing with the paper towel.  If some stain is being removed, keep turning the towel or rag to a clean area as you work. If this works, wipe the area with plain water when you are done.

If this does not work, you could try solvents, such as alcohol, mineral spirits, or acetone, but any of them are almost certain to compromise the finish. They also give off serious fumes and need to be handled with care.

Even the gentler detergent based cleansers may leave the paint with a different sheen from the rest of the wall. If the area is small and near trim, it may be worth dabbing a bit of matching paint over the stain. Depending on how long ago the wall was painted, it may blend and match well. If not, you may be facing a repaint of the section.