The artwork, said to have been valued at around £1 million (about NZ$2 million), first appeared in the coastal town of Dover in 2017, the year after Britain voted to leave the union in a controversial referendum.
The mural, which adorned the Castle Amusements building, depicted a worker chipping away at one of the 12 yellow stars on the blue EU flag.
Then, in 2019, the mural - which had become a tourist landmark in the town, a major gateway into the EU - disappeared overnight after being whitewashed.
Now, despite attempts at potentially restoring the work, the building on which it was painted has been demolished as part of a wider regeneration project.
Dubbed 'The Bench', the new development will comprise cultural and community engagement facilities, as well as residential dwellings, according to the local authorities.
In a statement sent to CNN, a spokesperson for Dover City Council (DDC) said: "Prior to authorising the demolition, and having taken professional conservation advice, DDC determined that the Banksy could not be viably conserved without considerable costs to local taxpayers, even if it were technically possible."
In a detailed explanation of the project online, the council said it had not been involved in painting or covering up the mural.
Nevertheless, the spokesman said that its contractor - DDS Demolition - is "attempting to see if the Banksy artwork can be conserved in any way."
He said there was "no certainty" about this "given the fact that the artwork was painted over and the poor state of the render," but added that the contractors had been "successful in removing stars and a section of the man and ladder intact."
The demolition company is meeting all additional costs which will mean that if successful, the artwork would be considered its property.
In an email sent to CNN, a spokeswoman for the demolition firm said that retrieved pieces of the mural are currently "being assessed."
She added: "DDS are making every effort to salvage the artwork for the community to enjoy."
The elusive artist, whose identity is still widely unknown, has not publicly commented on the demolition.
Though the market value of his work has soared in recent years, Banksy continues to paint images in public spaces that make preservation near impossible - and even invite theft or defacement.
'Spy booth', one of Banksy's most famous murals depicting three secret agents surrounding a phone booth with recording equipment, was destroyed in 2016.
And in February this year, an abandoned freezer forming part of an artwork that had appeared in the town of Margate was taken away just hours after Banksy confirmed the piece as his.
In 2018 the artist famously destroyed one of his own works when moments after selling at auction for US$1.4 million his painting of a girl with a red balloon passed through a shredder.