The problem has just been solved. The drain basket was solidly rusted
into the sink. I tried tapping it, I tried wedging the flange with a
screwdriver, I tried turning it with a large wrench, and I couldn't get
it to budge.
The local hardware store guy suggested using something called PB
("powerful penetrating catalyst"). I sprayed it around the flange in
the sink, and thirty minutes later, the basket lifted out.
I never thought something that said "As seen on TV" would work so
Many thanks for the responses, and I shall keep them in mind as I work
on other plumbing problems around the house.