Punch and Judy

Punch and Judy was a traditional puppet show.

After the Endovorms killed several civilians in Trafalgar Square, a Punch and Judy stand was abandoned, with the fleeing puppeteers still wearing the characters. (PROSE: The Lions of Trafalgar)

When the Sixth Doctor commented that Astrolabus was mad, he responded by hitting him repeatedly with a puppet he called "Mr. Punch". (COMIC: Voyager)

The Eleventh Doctor used a Punch puppet when he destroyed the Ice Governess with his sonic screwdriver, while exclaiming Mr. Punch's signature catchphrase, "That's the way to do it!" Later, the Ice Governess repeated this phrase several times while pursuing the Doctor. (TV: The Snowmen)

Extraterrestrial equivalents
When the Fifth Doctor was on Manussa trying to prevent the return of the Mara, Lon and Ambril observed a puppet show which strongly resembled a Punch and Judy show. (TV: ''Snakedance)

Behind the scenes
According to REF: The Brilliant Book 2011, a non-narrative source, when the Fourth Doctor was trying to get rid of a Cybermat, the young Winston Churchill thought it was part of the puppet show he was watching.