I've done some more playing around and, after my initial excitement, back button focus doesn't prevent the issue. That was just a fluke.
It's not user error. I've found that there's a display mode where you can see a green cross that shows what the camera (thinks it) focussed on. The green cross is consistently on the eye, whether I use eye decent or choose the focus point myself. But the nose or cheek is the part actually in focus. In one photo, the hand is in focus (several cm in front of the eye!). Green cross still on the eye.
I've tried with my 56mm f/1.2 - same problem. I'm incredibly disappointed and am thinking about trying to return the camera. These photos are irreplaceable and I'm really annoyed they're spoiled.