I think you need a photo or a better explanation. It sounds like you don't have a downspout or it's too short? If so, you need a proper one. If you have one, but it lets out by the house which is where the mud and run-off happens, then you should just dig a trench and attach the downspout to a flexible pipe and have it empty where you want it to. But it sounds to me like I must be completely mistaken. Of course, a decorative rain barrel would solve the run-off problem and be ecologically sound if you use the water for your flowers.

could you collect the water in a water butt and use it to irrigate specific planting areas maybe ??

get a gutter extender, they may not be the most attractive things ever but you do NOT want the spouts to dump all the rain water next to your foundation. Over time this will cause damage. Always choose infrastructure over aesthetics.

Can you put a collection of rocks to prevent the dirt from splashing? Or extend your downspout to drain somewhere else? Or collect in a rain barrel that you can use to otherwise water plants?

If you search for key terms "rain barrel" and "downspout" here I'm sure there will be lots of ideas. I saw a fantastic one on here that I thought I had bookmarked, but I guess I didn't. Boo :(

You can create a rain chain using buckets like the photo attached. You can also bury your downspout and run a pvc tube away from the home if you dont want any splashing.

In Switzerland, where I am, it is very common to add a chain and let the rain trickle down it... I think you need to experiment with the type of rain you regularly get (ie if you are in Florida during the summer season, no amount of chain is going to help you!!!)

Rain chains and downspouts will both do more harm than good if the water is not directed well outside the footprint of the foundation. Typically when we install rain chains, a collector box piped to a catchment drain system running to daylight (far from the house as we can get it) is installed. Downspouts are similarly hooked to an underground tiling that directs the water away from foundations. Allowing downspout water to collect next to the foundation is a source for erosion and damp in basements and crawlspaces.