I noticed that the wire into a ceiling lamp lit up an electrical
screwdriver when I checked its terminal, and it did the same at the
switch end - even when I completely disconnected it from both ends.

I'm pretty sure that this was a single uninterrupted wire from switch to
lamp, with nothing else connected to it, but I can't be 100% certain.

I assume that there was a very low induced current in the wire from some
other nearby wire; whatever it was, I couldn't actually feel anything
myself when I touched it.

Is this normal behaviour for an electrical screwdriver, to be so
sensitive?

Daniele