Okay, I found the culprit.
In December, I installed a Christmas theme for RStudio:
https://www.garrickadenbuie.com/blog/2018/12/05/holiday-rstudio-theme/
When I, as a test this morning, switched to the default (TextMate) RStudio theme, all performance problems went away!
So the theme apparently introduces some code that RStudio can't handle. Maybe the part of the Christmas theme that's to blame is a CSS animation (see at the bottom of the above linked blog post).
So, on the one hand, my problem is not stemming from RStudio, but on the other hand, this might be something that theme developers should be made aware of?
Anyway, my problem is fixed, it's a year until next Christmas, and I'll stick with the provided themes for the time being 
I would like to thank you for your swift and serious response!!
I also notify the theme developer.
Best,
Per