The wild world is closer than you think. We just don’t always see it. We don’t always look in the right places, or in the right way. Even wildlife enthusiasts can miss wildlife. It is easy to be unaware of the stuff that is going on all around.

Then, out of the blue, you can have one of those transcendent wildlife moments, one of those dramatic, thunderbolt revelations of the wildness that still occur, even in lowland Britain. Not even searching for it: sometimes the wild comes and seeks you out on some whim of its own.

It was just that way coming back from a family holiday in north Norfolk, me the observer rather than the driver. It took the form of a dramatic explosion along a not-especially-quiet country road, a bright red creature the size of a large goat crossing pell-mell in front of us, missing us by about ten yards or half a second, and hammering on into a field of ripening corn.

As front-seat passenger, I had the best view and a longer look besides. And it was unmistakable: coat the colour of a ripe conker, an almost-tailless pale bum, fine deep eyes, and a pair of little antlers like Twiglets. Roe deer! Roe buck, to be precise.

There are six species of deer in this country: remarkable when you think how big they are and how small the wild places we have left for them. They exist in hefty numbers as well but they are tremendously good at being invisible. Four of the six species are introduced: the fallow deer, big and spotty with elaborate antlers, came over with the Normans.

The other three all escaped from private collections and established themselves over the past couple of hundred years: muntjac, sika and Chinese water deer. The true Brits are red deer, the monarchs of the glen, and roe deer. But we had to start again with the roes. They were hunted to extinction by the beginning of the 19th century, and were reintroduced.

This time, in a less unrelenting hunting culture, they have managed to stick around: lurking at the edge of forests, eating leaves, shoots, berries and grass, seldom seen but existing in fine numbers. I can remember only two uncompromising sightings in the past ten years or so. One startled when I was looking for badgers, running in plain view along a hedgerow, its pale bum flashing out its warning to all other creatures in sight. The second was in plain view.

Animals have a flight distance. If you are outside it, you can observe each other without prejudice. A roe deer will stand motionless, a solemn-eyed gaze on the intruder, ready for a quick nought-to-sixty at the first hint of threat. This one, a fine little buck, I encountered when looking for water voles. Roe deer are not things you go looking for. You just bump into them. It’s pretty rare to have a good sighting of one but the deer themselves are not rare. That is their great paradox. The British population is somewhere between half a million and a million: a fair old margin for error. They have a genius for being overlooked.

They are still shot by those with a taste for such a thing. There are plenty of tales about roe deer munching in plain view of a party of chattering walkers in fluorescent cagoules, but vanishing into nothingness at the first hint of a stalker silent-footing through the woods in full camouflage.

Now here’s a thing: Bambi was once a roe deer. He was the hero of Bambi, a Life in the Woods, a novel by the Austrian writer Felix Salten.

Disney turned him into an anthropomorphic white-tailed deer. (Incidentally, the robin feathering her nest in the Mary Poppins film is an American robin, a unique example of this species nesting in London.) But it’s the vanishing that is so remarkable about roe deer. Most deer — at least, those who must live close to humanity — possess this ability to some extent, but with the roe deer, it is quite remarkable.

The Spanish call the roes el fantasma del bosque — the phantom of the woods.

Ted Hughes, not a man given to Disneyfication, got the hang of roe deer all right. Recalling an encounter with a roe or two in the snow, he said: “They had happened into my dimension.” And that’s what roes do. They live triumphantly separate lives, every now and then happening into our dimension.

The sight of any deer, but particularly a roe deer, is not so much an encounter as a blessing: a revelation that the world is still wilder than we think — wilder than we dare to think.

Deer are not universally loved, even by conservationists. Their taste for fresh and juicy growing things drives well-meaning people mad, if they are trying to encourage new growth in an old wood. Many a well-managed forest has areas of fencing to keep the deer out: exclosures. Woodland management requires deer management, which sometimes takes drastic form. In the 1960s, roe deer were seen as vermin by forestry interests; they are a little more relaxed now.

But roe deer live their secretive lives, and are not only very good at not being culled, they are even better at recovering from a cull, with a great propensity for twins. And occasionally they appear, cast a sudden piercing light of wildness into our tame lives, and then vanish again. After that, we must slip, as Hughes had it, “back to the ordinary”.