First of all, I must go back on my words - a hardware reset did cure the problem, but not for long! After a restore it was destroyed again.
Secondly, I took the unit for repair. They replaced the internal battery and the ROM. All looked OK until I restored my data and then the same problem. I then went and deleted all the installed programs, but it still had the problem. A hardware reset cured it again - and a restore took me back to square one.
I guess there is something corrupt in some register in the backed-up system, so I'll try downloading the programs one by one from the PC. What do you think?