The closest confirmation on this we have is Leaf telling Bran they do exist, however, we also have other sources claiming them to exist. It would appear as though they've fallen into the same realm as the White Walkers and giants with the majority believing them to be fairy tales but some still knowing the truth.
And they did sing. They sang in True Tongue, so Bran could not understand the words, but their voices were as pure as winter air. "Where are the rest of you?" Bran asked Leaf, once.
"Gone down into the earth," she answered. "Into the stones, into the trees. Before the First Men came all this land that you call Westeros was home to us, yet even in those days we were few. The gods gave us long lives but not great numbers, lest we overrun the world as deer will overrun a wood where there are no wolves to hunt them. That was in the dawn of days, when our sun was rising. Now it sinks, and this is our long dwindling. The giants are almost gone as well, they who were our bane and our brothers. The great lions of the western hills have been slain, the unicorns are all but gone, the mammoths down to a few hundred. The direwolves will outlast us all, but their time will come as well. In the world that men have made, there is no room for them, or us."
A Dance with Dragons, Bran III
It is claimed that Euron Greyjoy used unicorn horns to turn people to his cause, though it's never stated if these were real or not (more on this later).
Aye, he thought, a great victory for the Crow's Eye and his wizards. The other captains would shout his brother's name anew when the tidings reached Oakenshield. Euron had seduced them with his glib tongue and smiling eye and bound them to his cause with the plunder of half a hundred distant lands; gold and silver, ornate armor, curved swords with gilded pommels, daggers of Valyrian steel, striped tiger pelts and the skins of spotted cats, jade manticores and ancient Valyrian sphinxes, chests of nutmeg, cloves, and saffron, ivory tusks and the horns of unicorns, green and orange and yellow feathers from the Summer Sea, bolts of fine silk and shimmering samite... and yet all that was little and less, compared to this.
A Feast for Crows, The Reaver
Davos himself calls Skagos "the isle of unicorns and cannibals" and Sam reads that unicorns live there with Dareon even commenting that they might see one.
Nine-and-twenty ships had set sail from the Wall. If half of them were still afloat, Davos would be shocked. Black skies, bitter winds, and lashing rains had hounded them all the way down the coast. The galleys Oledo and Old Mother's Son had been driven onto the rocks of Skagos, the isle of unicorns and cannibals where even the Blind Bastard had feared to land; the great cog Saathos Saan had foundered off the Grey Cliffs.
A Dance with Dragons, Davos I
The island sat at the mouth of the Bay of Seals, massive and mountainous, a stark and forbidding land peopled by savages. They lived in caves and grim mountain fastnesses, Sam had read, and rode great shaggy unicorns to war. Skagos meant "stone" in the Old Tongue. The Skagosi named themselves the stoneborn, but their fellow northmen called them Skaggs and liked them little. Only a hundred years ago Skagos had risen in rebellion. Their revolt had taken years to quell and claimed the life of the Lord of Winterfell and hundreds of his sworn swords. Some songs said the Skaggs were cannibals; supposedly their warriors ate the hearts and livers of the men they slew. In ancient days, the Skagosi had sailed to the nearby isle of Skane, seized its women, slaughtered its men, and ate them on a pebbled beach in a feast that lasted for a fortnight. Skane remained unpeopled to this day.
Dareon knew the songs as well. When the bleak grey peaks of Skagos rose up from the sea, he joined Sam at Blackbird's prow, and said, "If the gods are good, we may catch a glimpse of a unicorn."
A Feast for Crows, Samwell II
The World of Ice and Fire also comments the same as Sam reads about the unicorns.
The Skagosi who reside there are little regarded by the other Northmen, who consider them no better than wildlings and name them Skaggs. The Skagosi call themselves the stoneborn, referring to the fact that Skagos means "stone" in the Old Tongue. A huge, hairy, foul-smelling folk (some maesters believe the Skagosi to have a strong admixture of Ibbenese blood; others suggest that they may be descended from giants), clad in skins and furs and untanned hides, and said to ride on unicorns, the Skagosi are the subject of many a dark rumor.
The World of Ice and Fire, The North: The Stoneborn of Skagos
However, not all the maesters appear to believe that the unicorns were real, though that appears to descend from disreputable merchants selling fake unicorn horns. Some unicorn horns have been seen that appear to be real or at least different to anything else seen before. Although the maesters are yet to find a living example or skeleton.
The "unicorns" of Skagos were once scoffed at by maesters at the Citadel. The occasional "unicorn horn" offered by disreputable merchants has never been more than the horn of a kind of whale hunted by the whalers of Ib. However, horns of quite a different kind—reputed to be from Skagos—have been seen by the maesters at Eastwatch upon occasion. It is also said that those seafarers brave enough to trade on Skagos have glimpsed the stoneborn lords riding great, shaggy, horned beasts, monstrous mounts so sure-footed they have been known to climb the sides of mountains. A living example of such a creature—or even a skeleton—has long been sought for study, but none has ever been brought to Oldtown.
The World of Ice and Fire, The North: The Stoneborn of Skagos
Lastly, some claim unicorns can be found in Ib which seems to agree with unicorns from Skagos as some maesters have claimed that the Skagosi have a strong admixture of Ibbenese blood.
Ib is the second largest island in the known world; only Great Moraq, between the Jade and Summer Seas, is larger. Stony and mountainous, Ib is a land of great grey mountains, ancient forests, and rushing rivers, its dark interior a haunt of bears and wolves. Giants once dwelt on Ib, we are told, but none remain—though mammoths still roam the island's plains and hills, and in the higher mountains, some claim unicorns can be found.
The World of Ice and Fire, Beyond the Free Cities: Ib
Note that the purple unicorn sigil belongs to House Brax and that they are said to have been formed from marriage between the First Men and the Andals. As an historical house it is likely that more unicorns were around when it was created and so hence the sigil.
A party of mounted horsemen rode forward to challenge them as they approached the stakes. The knight who led them wore silver armor inlaid with amethysts and a striped purple-and-silver cloak. His shield bore a unicorn sigil, and a spiral horn two feet long jutted up from the brow of his horsehead helm. Tyrion reined up to greet him. "Ser Flement."
Ser Flement Brax lifted his visor. "Tyrion," he said in astonishment. "My lord, we all feared you dead, or…" He looked at the clansmen uncertainly. "These… companions of yours…"
A Game of Thrones, Tyrion VII
In time, Lannister kings wed their children to Andals as well; indeed, when Gerold III died without male issues, a council crowned his only daughter's husband, Ser Joffery Lydden, who took the Lannister name and became the first Andal to rule the Rock. Other noble houses were also born in such unions—such as Jast, Lefford, Parren, Droxe, Marbrand, Braxe, Serrett, Sarsfield, and Kyndall. And thus revitalized, the Kings of the Rock expanded their realm still farther.
The World of Ice and Fire, The Westerlands