Oh no MD your poor ponies! I'm sure someone has helpful ideas to remove it though so you don't have to get rid of your girls. :(
Yes as breyer says its really just dirt - you are best to try your hardest to remove it - which is possible with determination [there is an easy tutorial in the sticky at the top of the Corral] - as when removed it can leave little depressions in the plastic which may get bigger if the dirt is allowed to build up further.
I always think a better description of pindot is "in grained dirt" as plain and simple as that :)