Sony has released firmware update v1.10 for the Alpha 1, which, according to the manufacturer, brings some minor improvements, including AF performance and stability, as well as "eye sensor detection performance in strong light sources such as sunlight". However, the update also appears to fix an in-body stabilization bug that some users occasionally encountered.
According to this, the IBIS of some A1 cameras turned on in some situations only several seconds after the start of the recording, sometimes it took up to 10 seconds before the image was stabilized. The problem seemed to occur mainly when the camera was moved shortly before the shot was taken, regardless of the optics used (though with unstabilized lenses). The following video demonstrates the problem, which as the comments show also occurred with some other buyers, while others could not reproduce it.
Without explicitly pointing it out, the bug has now apparently been fixed by Sony with the new firmware v1.10.