Britain was on a knife edge when Lucy Connolly, a mother and former nanny, tapped out an inflammatory 51-word post on social media a year ago, calling for “all” hotels that house asylum seekers to be set on fire. “If that makes me a racist, so be it,” she added, a postscript that did nothing to spare her the wrath of the country’s courts.

Her post was viewed 310,000 times. Now serving a 31-month prison sentence for inciting racial hatred, Ms. Connolly has become a charged symbol in a debate over whether Britain is suppressing free speech. Last month, an appeals court refused to reduce her sentence, drawing angry protests from her defenders, mainly on the political right.

Ms. Connolly’s arrest came during a fevered period after the killing of three young girls by a British son of Rwandan migrants, which ignited anti-immigrant riots across the country. She is one of thousands of people arrested in Britain each year for sending or posting offensive messages, which can include those that are “indecent or grossly offensive” if they are intended to cause distress to the recipient.

Critics, including Vice President JD Vance and Elon Musk, accuse Britain of backsliding in the age of social media. They point to a thicket of laws, some old, some new, combined with overly zealous policing, and a perception, on the right and the left, that law enforcement is biased against their points of view.

Britain, they say, is restricting speech in ways that affect “not just the British,” as Mr. Vance put it in February in an Oval Office meeting with Prime Minister Keir Starmer, “but also affect American technology companies and, by extension, American citizens.”

Mr. Starmer gives no ground when asked to respond to Mr. Vance’s claim that “free speech, I fear, is in retreat.”