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The 76-year-old actress said receiving her damehood from the Princess Royal was “thrilling” and “like a fairy story”.
The actress and campaigner, known for playing Patsy in Absolutely Fabulous, was given the award for services to drama, entertainment and charitable causes.
When asked what the honour meant, Dame Joanna told the PA news agency: “It’s huge. It’s absolutely huge.
“I mean these are things that, as a child being ticked off and having my prefect badge taken away for smoking, something that you would never dream of is that you would be honoured in Buckingham Palace to become a Dame.”
She added: “[Anne is] just such a wonderful sort of figurehead for women in this country, I think [she is] courageous and active and modest and brilliant.”
She said the investiture ceremony with Anne was “very different” to when she collected her OBE in 1995.
She told PA: “It was in the throne room, which is much smaller and much more intimate.
“And instead of an actual military band, playing very softly up there was this beautiful bass and string quartet and it was so charming and so moving.”
Dame Joanna said she and Anne were both supporters of Farms For City Children.
She said Anne was “so lovely” and told her that the honour was “richly deserved”.
She said the future of television was “still going strong” and added she plans to continue working until they “hack the legs away from underneath me”.
Dame Joanna, who has a new travel series starting next month, said it will be clear when it is time to stop because “the phone will stop ringing”.
Professor Kathleen Stock, former professor of philosophy at the University of Sussex, was made an OBE for services to higher education at the same event.
Prof Stock quit the University of Sussex last year amid a row over her views on gender identity.
When asked if she would work in a British university again, she replied: “No, I think probably not now.”
Her decision came after an anonymous group, reportedly set up by students, launched a campaign to get her sacked amid accusations of transphobia.