I've been close before. Once a deer jumping out of the bushes in front of me, and I may be exaggerating the closeness of the encounter since it was many years ago, but I was travelling quite slowly on a trail and the deer actually passed over my front wheel.
I also was a couple meters from hitting a moose at a much higher rate of speed on another occasion. The moose sauntered out of the way just in time, luckily for me. The moose would obviously not have been injured, but it would have been at least like hitting a brick wall, if not hitting a brick wall and then being trampled.