The technician came over today and exchanged the motherboard. Actually, it's in two parts (CPU and I/O?); he exchanged the I/O part and the error still occurred, so the fault was probably on the CPU part, but he didn't have the right replacement part.
Fortunately, HP has a spare parts store in Hamburg, so the right board arrived shortly afterwards by taxi. The technician installed it, the machine booted... and died. When he removed the board, he found that a screw had got onto the board somehow (but not of a type he had been using), which had caused a short circuit or something. He marked it DOA (dead on arrival), so we didn't have to pay for it, and ordered another one. Fortunately, the store still had another board of the right type. After installing that, the machine booted properly.
(It thought it was still 2 January, the date it died, but Ralf cured it of that.)