OK, I think that sort of qualifies because you have a living bit of clothing that's important to the plot, and it's the fifth answer so the ball is in your court.
Regarding my hints (and the one I mentioned much earlier)
The TV show that had an episode about an invisibility suit was Castle, I forget the episode title. It's a crime show, but they've had several episodes with SF themes including one instance of time travel which appeared to be genuine, unless it was later explained away in another episode I missed.
1970s British novel about an alien world where fashion is integral to social status
https://en.wikipedia.org/wiki/The_Garments_of_Caean
A 21st century Japanese anime television series in part inspired by the above
https://en.wikipedia.org/wiki/Kill_la_Kill
An episode of a 1960s TV SF series set in the 21st century - the main protagonist of the episode (usually appearing in more secondary roles in the
series) is an aristocratic supermodel, and this profession is integral to the plot of the episode.
https://thunderbirds.fandom.com/wiki/Al ... ckenbacker
(Actually I was remembering this one slightly wrong - Brains is probably as important to the plot as Lady Penelope, although it does revolve around a revolutionary new fabric and attempts to steal the formula.)