Just to be complete a few more things that might be noteworthy and might give some people ideas about what was causing this:
* I was running Battlefield 1 with a normal Windows user, not an administrator.
* The Battlefield 1 update also got intalled while running it with this non-administrator user.
* This happened using a German language Windows install with *everything* switched to English (including the Windows default language). Battlefield also got installed using English as language.
* Another normal Windows user on this machine (my daughter) has a different Origin account with different games. In other words, there are multiple different Origin accounts being used from the same machine, but of course not concurrently.
* The machine is a Lenovo IdeaCentre Y910, which means the game was running in 1440p at 144Hz using DX11.
For now I've kept things as simple as possible: I installed Battlefield 1 with administrator user and I'm also running it with that user.