Our Capital CIty Trail has some pretty deeply wooded sections; I wouldn't be surprised if cyclists encounter deer crossing in front of them. I was just thinking about starting a "close-call animal encounters" thread. Ones that come to mind:
* Dead-end country road, road bike, aprx 20 MPH, slight descent, turned my head to notice a cow who had somehow wandered to "this" side of her pasture fence, soon as I was eyes forward again, a woodchuck is crossing the road in front of me. Bear-claw both brakes, and my front wheel misses his tail by about six inches.
* Rail-trail on a tandem, my buddy Adam stoking, a chipmunk crosses BETWEEN the front and rear wheels.
* Snapping turtle on a paved trail. Big enough that he would have spilled anyone not seeing him and hitting him. I stop, hop off, figure I'll use my frame pump to nudge him off the trail. I start bumping him in the back end of his shell, and he turns around and starts biting my pump, and looking REAL p!$$ed. Several attempts, same results, he gets madder each time. So I start using the pump to taunt him instead. Got him to start attacking it, and "led" him off into the grass.

One does see dead chipmunks on the in-town bike paths here, Besides them, the most likely animal encounter here is with the squirrels who see fit to reverse direction halfway while crossing the street.