If you drop the soap in like mud or dirt or some shit then yes, it will get dirty. Best dumb answer award goes to me.
My synopsis on this is quite simple. Ever showered after a long day of working covered in just working persons filth? Because I have. And when you lather the soap and soap up your body you then put the soap down to rinse off. Therefore the soap still has suds on it. But wait, The suds sometimes have the dirtyness from your body transfered to the soap. So that means the soap has dirty soapy suds that are clearly visible on the soap and if you are like me I always rinse off my soap until there are no visible suds on the soap because I like my soap clean. Sometimes I'll see soap where they lathered and the soap dried with the dirty soapy suds on it leaving dirty residue on the soap. So yes. Soap can become dirty if not rinsed off using itself to clean. Think of a tool that is "self cleaning" after every use. Eventually the tool that self cleans will need to be cleaned itself because it is infact dirty. But there is also bodywash which I guess is also a soap. But it's in a bottle.. never seen a bottled soap get dirty.