OK friends, I have been wanting to follow this issue through and try to contribute a good solution.
I did research, as mdsalemi suggested, on cleaning vinyl, and a bunch of product names came up which I do not have access to.
However, "baking soda" kept coming up, and I have not tried that for one reason.
You need to mix it with water until it becomes like a paste. This would be fine to use on solid surfaces, but not like the headliner, which is full of tiny holes. I would end up with a lot of the paste under the liner which I will never be able to remove.
Someone suggested vinegar, so I will try that out.
Someone else suggested WD40. again I will try that out too and keep you posted.
On the other hand, the small area which I tried with petrol looks ok and still dry.
Just now whilst writing this, I remember having encountered this situation with my multi chested tool box.
Some of those tools were never used for years, and about ten years ago, whilst cleaning the box out, I noticed the same thing with their plastic/rubbery handles.
I cleaned them then with petrol which seemed the easiest solution, and to date I have not experienced more degeneration of the sticky stuff.
So, maybe we are onto an easy solution there???
Till my next report
Cheers