The Broadcast Advertising Clearance Centre has stopped Madasafish from screening the campaign, in which Enfield reprises the Frank Doberman character, believing that British people will not want to see their national heroes subjected to hectoring.
The spot shows yob Doberman shouting at an actor portraying the wartime Prime Minister Churchill. Enfield tells him he should be using Madasafish if he wants to download naughty pictures of Britain's most famous Second World War military leader, Field Marshall Montgomery.
"Oi, Churchill! Well done for winning World War Two, nice one. But if you were downloading saucy pictures of Monty from El Alamein using a dial-up connection, I should say 'Oi, Churchill, no! You should be Madasafish, you porky Prime Minister'," Doberman shouts.
The ad promotes Madasafish's broadband internet services and is one of a series of three. Captain Cook and Albert Einstein lookalikes appear in the other two ads.
The campaign was created by Enfield, who has a business deal with Madasafish that will see him receive an income from new sign-ups generated by the campaign.
Enfield hit out at the ban, describing it as ridiculous. He said: "I think people who are offended by it would either have to be without a sense of humour or a little bit bonkers."
The ISP planned to retaliate with another ad, where Enfield talks to camera about the ban, telling people where they can watch the spot online, but this ad has also been banned from broadcast, on the grounds that it refers to a previously banned ad.
According to the Sunday Telegraph, Churchill's descendants, including Emma Soames, are in favour of the ban. However, historian David Starkey has called the ban "preposterous".
David Laurie, chief executive of Madasafish.com, said: "The BACC said it had been banned because it could cause offence because Churchill is one of the most revered Britons in our history. We think it's absurd that BACC regard this as offensive."
He added: "We have the utmost respect for Churchill and he really was one of our greatest heroes. We would never want to do anything that would cause upset but this country does have a tradition of gentle mockery and irony, which is a quintessential British trait."
If you have an opinion on this or any other issue raised on Brand Republic, join the debate in the .