The question is triggered by the wonderful animations by Jason Hise:

https://www.youtube.com/watch?v=LLw3BaliDUQ

https://www.youtube.com/watch?v=6Ul_-ABYaYU

https://www.youtube.com/watch?v=aYVt1UiERIQ

All these animations are based on the well-known belt trick (a way to represent SU(2) as double cover of SO(3)) and suggest that a solid sphere that is glued into a soft mattress can rotate continuously, if the region of mattress close to the sphere performs certain appropriate motions around the sphere. Is this conclusion correct? Are there papers on this very counter-intuitive issue?

The belt trick is shown impressively in https://youtu.be/DHFdBWU36eY . It is also called the plate trick, the string strick, the balinese dance trick or the balinese candle trick. The trick shows that an object connected to spatial infinity can rotate indefinitely, coming back to its original position after a rotation by 4 pi = 720 degrees.

It is not evident that the belt trick implies that a sphere can rotate when glued in a mattress. Many discrete belts do not make a continuous and smooth mattress. So it could be that the mattress property does not follow from the belt trick - it could be that the belt trick requires discontinuities. So the question is: is the mattress trick a continuous operation on the mattress, or does it require cuts in the mattress?

Did anybody ever perform the experiment in real life? For two dimensions, this has been done, as shown in this video: https://www.youtube.com/watch?v=UtdljdoFAwg that shows that a ball glued into a handkerchief can be rotated continuously. But it is really possible also for three dimensions, for a full mattress?

The issue is interesting because a ball in a mattress can be seen, if this works, as a model for a spin 1/2 particle. Above all, if we assume that the mattress is a model for space itself, the ball in the mattress would be a way to model a spin 1/2 particle with the help of a smooth 3-dimensional manifold, something which is often assumed to be impossible. It would solve one of the contradictions between general relativity and quantum theory.

The newest animation by Jason Hise of "Dirac's handkerchief", made after this question was posted, is here: https://youtu.be/tazjVJcxm50 It realizes the answer of André Henriques. The answer to the original question is thus a clear "yes".

If anybody every manages to do this in an experiment, I'd like to see the video -- and to put it into my free physics text!