Yes, it turned out to be the power supply - or rather, strictly speaking, the power cable. I'm using a generic Maplins mains/USB charger-adaptor (and I also tried an Apple charger), linked through a perfectly functional mobile 'phone USB/Micro-USB connector cable. That gave me the problem described....

(Oh yeah - tried two different power supplies, too. The orange light is on, solidly, in all cases.)

I've hooked up my brand new RP to an HDMI TV, plugged in an SD card with Raspbian, and switched on. The Power light comes on orange. I get the rainbow splash screen, then a blank screen, then a screen full of text scrolling up too fast to read... Then the spash screen, and the process starts again, ...