Sometimes while I'm writing some code, a small popup appears at the bottom of the editor (covering the search box, if it's open). The Tips & Tricks page lead me to think it's called Assistant.

I find it very unhandy because of it covering the search box: sometimes it pops up while I'm using the "Next" button of the search box and I accidentally click on one of the options it proposes (and after days I find out the definition of a member function I never did nor meant to do...)
Is it possible to disable this feature without turning off autocompletion?

No, currently it is not possible to turn the assistant off.

Greetings