About 250 are thought to exist in the wild, putting the species at risk of low genetic diversity and inbreeding.
But new research suggests that the lynx has had little genetic variability over the last 50,000 years, and this has not hampered its long-term survival.
The study is published in the journal Molecular Ecology.
The authors say that the findings should offer hope to conservationists who are trying to pull this critically endangered cat back from the brink of extinction.
Love Dalen, from the Swedish Museum of Natural History, said: "This indicates that some species can do fairly well at low population sizes, even for a very long period of time."Last year, science reporter Rebecca Morelle and Guillermo Lopez, from Lynx Life, managed to get a rare glimpse of a wild Iberian lynx
The Iberian lynx was once widespread across the Iberian Peninsula, but hunting, habitat destruction and a disease that nearly wiped out its main source of food - rabbits - contributed to its crash in numbers.
From the 1960s its numbers plummeted, dropping from an estimated 3,000 to approximately 150 in 2005. The lynx is now only found in two isolated pockets in the south of Spain.
Conservationists, such as the Lynx Life team, who have been successful in boosting lynx numbers through captive-breeding programmes and efforts to increase rabbit populations, are concerned about low genetic diversity in the cats and the problems this could cause.To study this further, a team from Spain, the UK and Sweden, looked at the mitochondrial DNA - a part of the genome that is usually extremely variable - of several lynx, and found that there was, as expected, little genetic variation.
But when they compared this with mitochondrial DNA from lynx fossils spanning the last 50,000 years, they found that there was hardly any variation between the ancient specimens and in the modern-day animals.
Professor Mark Thomas, an author of the paper from University College London, said: "This is the first species, as far as I am aware, where such low diversity has been seen recently and over such a long period of time."
Usually, he said, animals alive today that have a low genetic diversity had a much higher level of genetic variability thousands of years ago.
He explained: "It takes a while for a bottleneck to get rid of genetic variations.
"So usually, when people see low variations today, they say that's probably because of big events in recent evolutionary history, in the last 10,000 in particular, where the climate got much warmer after the Ice Age, humans became more and more dominant because of farming etc.
"That reduced these populations and therefore squashed out the genetic diversity."
However, the Iberian lynx was different, and its consistently low genetic diversity suggested that it had always had a small, long-term population size, Professor Thomas added.
The researchers are not sure exactly how the Iberian lynx has managed to survive with this over thousands of years, while other animals have not.
Dr Dalen, also an author of the paper, said: "This type of low diversity has not really been observed in any other species.The researchers compared the DNA from modern-day cats with that of ancient animals
"From a conservation point of view this is very interesting."
Usually, he said, this would lead to inbreeding and a reduced potential to adapt to changes in the environment.
But, he added: "There is a difference between being inbred and suffering from it.
"It seems highly likely that the lynx are inbred. In most cases, it seems, that inbreeding leads to a decreases in fitness - genetic problems such as deformities, or lower litter sizes. But there is a possibility that some species can handle that.
"One reason is that they simply don't carry that many bad genes; the other is that natural selection may eventually purge it from the population."
Dr Cristina Valdiosera from the University of Copenhagen added that the paper suggested that a lack of genetic diversity in an endangered species should not hamper conservation efforts.
She said: "It's a myth that certain species are doomed by their genetics.
"If a species is doomed, it is only doomed by a lack of will to conserve it."