Page template not applied

I want to use a different template for my Pages.
fair enough, I created a new one and found it under “Page Template” in the “Write Page” menu.
Selected it, saved the page and waited for a change.
Nothing happened. Reloading didn’t help.
Suspecting a cache problem, I looked at the page in another browser. Same result.
The page clearly uses the “index.php” template.
Even if there was a problem with the new template, I’d have expected it to default to “page.php”, which is present in the active theme.
Am I overlooking something simple? I recently upgraded to 2.o.6 but didn’t need the page functionality before, so I can’t tell whether this might be an upgrade problem.