The tiny dachshund puppy was born in Seoul, South Korea, at the end of last month following a competition advertised in the UK offering the procedure free of charge.
The dog was copied from a 12-year-old pet called Winnie, owned by Rebecca Smith, a cook from west London.
"The world will be a better place with more Winnies in it," Ms Smith told a Channel 4 documentary to be broadcast tonight. "Everyone who meets her loves her."
Ms Smith, 29, told the programme she acquired Winnie when she was 18 and the pet had helped her overcome bulimia.
The company that carried out the procedure, Sooam Biotech, has already created more than 500 cloned canines for owners around the world, but "mini Winnie" is thought to be the first British dog to be cloned.
Dolly the Sheep, the first mammal to be cloned from an adult, was created in 1996 by researchers at the Roslin Institute in Edinburgh.
Professor Sir Ian Wilmut, who led the Dolly team, told the programme he believed owners might be disappointed by dog cloning. "So much of the personality of a dog probably comes from the way that you treat it," he said.
The £60,000 Puppy: Cloning Man's Best Friend will be shown on Channel 4 at 10pm tonight.