Magic Mouse 2 behaves strange

Since a couple of weeks ago, the arrow on the screen would - for most of the time - stop changing to that pointing finger which it is supposed to do, when moving over a link, a list of options or whatever. In this way, I will for the most part only get a menu (like "Back", "Save as" and so on) instead of opening the link. Sometimes it will just mark a word in the text instead of opening the link.
Battery capacity is high, the iMac is about 9 months old, there is no dust on the mouse, and I've done several restarts without result.

This is VERY annoying when I'm using Facebook, which often takes a bit of clicking here and there.

Any ideas what I should try next?