According to Lars :
It has a two-core mains
lead.

Does it have an earth (ground) prong too?

Today I touched the outside of the lampshade and got a sort of vey
mild tingle feeling which felt "odd". When I used a mains tester
screwdriver on the exposed metal (at the joint of the lampshade
and support arm) then it glowed as if the metal of the Anglepoise
lamp was live.

If you get a continuous tingle, then it's _highly_ unlikely to
be a capacitive/inductive/static leak (as others have suggested).

Last time I touched a device and got a minor tingle, repeating
that test while holding on to something grounded was something
I'm not going to repeat. That hurt! Dumb me. At least
we're only 120V...

You say "old", and the picture shows that the wiring in the
lamp is being moved whenever you adjust the lamp. Possibly
abraded and the conductors may now be intermittently contacting
the frame.

I recommend removing the wiring and inspecting both it and
the lamp socket as others have suggested. The socket may need
cleaning. If you see worn spots in the wiring, replace it.

If you don't see any obvious worn spots, now you really worry
about the fixture.

Measuring resistance won't be particularly reliable, fancy
equipment or not, because if you have a bare spot on the wiring
in the arms, the slightest bit of movement may break/make the
connection.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.