My friend asked me if a 150-200 km wide island (more precisely, a lithospheric plate of such size) that rotates insanely fast by the geological terms, something around one rotation per week, is possible. I don't think it is possible via ordinary physics though, since the energy needed for that would likely just melt the whole surroundings and vaporize the seas. But can this exist with some handwaves or artificial solutions?
Made out of rock, check.
200km wide, check.
Rotating once per week... umm... Would you settle for rotating every 2.3 days?
All you need is some floating rocks, and a natural gyre in the ocean.
If you want even more solid rock, as in an actual tectonic plate spinning around, you are out of luck.
Tectonic plates as a rule do not engage in rotating motions, their primary movement is linear, slamming into each other or pulling apart or rubbing along each other.
All three of these activities tend to be accompanied by plenty of earthquakes, very often volcanoes, and very little movement. Even the most hasty of tectonic movement, such as the mid-Atlantic rift opening up, proceeds at a rate of less than 5 cm per year.