Cat Playing on slippery surface, worried about injury possibility.

Hi all, Hope you are having a good day.
I've moved into a new place that already had a beautiful pure white cat (that molts like no tomorrow but that's another issue). I've been playing with her using my bath robe tie like string(planning to get better toy) and she loves it. I can even get her jumping quite high after it. What I am worried about is my flatmate, who owns the cat, also has a Cow skin rug in the lounge where most of the playing occurs. This cow skin is Very slippery and I have noticed the cat can and will slip on it if running too fast or landing an awkward jump. Should I move our play sessions to a different area so as to not cause an injury if she lands wrong and falls over? So far its been hard to tell if shes landed wrong as she naturally likes to flip onto her side once having caught the bath robe tie so she can get at it with her back legs.
Thanks for reading

 Welcome to TCS! I would definitely monitor the cat's behavior after one of those slips/falls and also ask the owner. Taking a video to show to your vet might be a good idea as well.

While most of the time, my cats have not been harmed in similar situations, I had a kitten who, during rough play with littermates (they were various ages as they were former ferals that I had captured) got flipped off of the woodburner-hearth and landed on Pergot flooring and his head got tilted. My vet said that nothing could be done but wait and see as the Xrays didn't show anything but that didn't mean that there wasn't soft tissue damage. The cat grew up and his head eventually became less tilted, other than that head tilting, he is very healthy years later. However, I myself have neck damage and I can't help but think that he probably has some sort of pain and how I wish that the accident had never happened.