Don't despair just yet.
I got my old Sony PRS 500 out from the box where it had been reposing for the last six months, opened the cover, and saw a very similar picture.
"Oh Noooo!," I thought. "It must be a broken screen substrate."
However, I tried a long mains recharge (overnight) and the pattern vanished. The device is now working as well as ever it did.