Geronimo has twice tested positive for bovine tuberculosis, and the Department of Food, Environment and Rural Affairs (Defra) has ordered he be euthanised.
His owner, Helen Macdonald, who imported him from New Zealand, believes the tests are returning false positives, but has been refused permission to have him tested a third time.
Last week, Ms Macdonald lost her final appeal to save her beloved pet at the High Court in London and now a warrant has been signed for his destruction.
But she has received an outpouring of support from the public, with nearly 100,000 people signing a petition calling on Boris Johnson to halt the killing.
As around 30 protesters began a march from Defra's headquarters in Smith Square, Westminster, to the gates of Downing Street, the Prime Minister's official spokesman said: "We know how distressing losing animals to TB is for anyone. That is why the Environment Secretary has looked at this extremely carefully and interrogated all the evidence.
"The fact remains that Geronimo has sadly tested positive twice using a highly specific and reliable and validated test.
"This is something the Environment Secretary has looked at very carefully."
The demonstration has been organised by members of the Born Free Foundation, the Alpaca Society, and practising vet and bovine TB policy expert Dr Iain McGill.
The campaigners believe that Geronimo is free of TB and that Defra's tests are highly likely to be inaccurate.
They are demanding a different type of test be used to prove Geronimo's disease status before his death.
Speaking from her farm in South Gloucestershire, Ms Macdonald criticised the Government for refusing to change its mind.
"Unfortunately they are still misquoting data," she said.
"What they did to Geronimo was not a validated test - they knew what they were doing.
"I do not want to be singled out. It's about making decisions that have backfired on them and taking it out on Geronimo and causing him harm and me and my family harm purely because they didn't listen to us four years ago when they thought there was a problem.
"We are just asking to have him tested with something appropriate. I get they have policy to follow but there are other ways, and they don't have to kill him. He is safe in isolation here.
"They've always been happy with that and he's not a public health risk. They won't test his friends, so they are obviously not worried that he is going to give them TB.
"No-one has died here from TB in four years, so I just don't understand why it has to be this drastic."
Ms Macdonald said that when Defra officials do attend her farm to euthanise Geronimo she will not break the law.
"I'm not going to break the law but I am not going to make it easy. I'm not going to be helping them kill an animal that does not need to be killed," she said.
"This is an animal injustice and everyone can relate to it and people are coming from all sectors of farming to support us.
"We can make a difference. Geronimo is the poster boy for doing a better job than what we have done in the past."
At her farm in Wickwar, friends, family and supporters have joined her to protest against Geronimo's impending fate.
"We had a call to action for people to come to the farm and the support has been absolutely fantastic," she said.
"No-one wants to see anything happen to Geronimo and there must be a way forward and we just need to sit down and work things out and do the right thing for everybody."
As well as alpacas, badgers have been a victim of the fight against bovine TB, with mass culling employed to stop the spread since 2013, sparking a huge public backlash.
Campaigners are calling for the Prime Minister to force Environment Secretary George Eustice to halt the killing and immediately implement the latest bovine TB tests for all suspected cases.
They also want to see an end to all further badger cull licences in favour of a vaccination programme for cattle, alpacas and badgers.
The outcry over Geronimo's fate prompted Mr Eustice, who comes from a farming background, to write an article in the Mail on Sunday about his own experiences with bovine TB.
"Each week on average, we have to remove more than 500 cattle from herds due to infection in England alone. Behind every one of those cases is a farmer who has suffered loss and tragedy," he said.
"Farmers understand that infected animals are a risk to the remainder of their herd, so, while the loss of individual animals is always a tragedy, the farming communities have worked with our Government vets in this arduous but necessary endeavour."
Ms Macdonald has threatened to film the last moments of her alpaca's life if the cull goes ahead, and broadcast it on social media.
Asked about the intervention by the Prime Minister's father, Stanley Johnson, to save Geronimo, Business Secretary Kwasi Kwarteng told Sky News: "I'm not sure where Stanley's coming from on this.
"My understanding is that the alpaca was tested positive for TB and the rules are that they have to be culled, because bovine TB is really, really damaging to farmers and people who make their livelihoods in agriculture and that's why we have the policy."
He added: "I don't think it's a PR disaster; I think, as you say, it's an August story. It's obviously difficult because there's a lot of people invested emotionally in a story of an animal, but there's a policy and there's no reason why the policy shouldn't be stuck to."