You sound like my old supervisor! It's pixel number vs counts, I've added an updated plot. Note again that as the CCD is returning a low signal for saturation, zero photons = high signal.
I'm out of the lab, it being the weekend, but I'll try to get some photos on Monday.
The setup is the reference circuit from the datasheet, 5V in, ground and data out. The chip is specced from 0-5.5V. As mentioned I thought the power supply was regulated, but it looks like it probably isn't. It's coming (via a Raspberry Pi) from a bog standard 5V wall supply which is probably around 5% accurate and I guess has lots of switching noise on top of it.
I'm fairly sure that's the culprit, but I was wondering if anyone had had similar experiences with CCDs and noisy power.
I'll see if I can get hold of a scope trace, I'll disconnect the ADC side. I don't see why it would make a difference as the signal level is generated in the CCD, a DC offset caused by a bad analog frontend would always be present, not just with long exposures.