The last time I was on Mull, there was a young chap skipping from stone to stone in a rather fast stream holding a Canon EOS 1D (not sure as to the model) held in one hand with no strap. I suppose that if he fell in, the camera would be shot, strap or no, but losing his footing and not falling in would have put his camera at grave risk.
I can drop anything, anytime, so I always use a strap unless the camera has no strap lugs in which case it needs to fit in my pocket.