Have had deer run in the field along side me then cut across the road in front of me. Well in front so no concern of being run over.
One of my other animal encounters was with a groundhog that I thought was dead. As I came up on it I was going to go between it and the edge of the road. I was quite surprised when it darted for the ditch and both wheels of my bike wheels went directly over him. No harm to either of us that I know of.
One other time I came up behind a couple of younger Amish boys with horse and buggy. As I neared they kept looking back and I could hear the horses gallop getting faster and faster. As my pace increased to catch them they kept looking, laughing and the sound of the horse shoes hitting the pavement continued to increase. I think by the time they turned into a drive, we were around the 18 - 19 mph range. Quite fun for both I think. Most older Amish just watch as you go around.