I think the bottom line is, it depends on how much thinking time you have. I would accept that *if* I were in the habit of swerving onto the pavement whenever a mouse ran across the road, I'd be a menace to other road users!
I can imagine a situation where a driver is confronted with an animal on the road and does not have sufficient thinking time to make the right decision.
Here's a case from ages ago, but needless to say it is decided only on its own facts, not as a binding precedent on all similar cases:
Ritchie's Car Hire v Bailey The plaintiffs had hired a car to the defendant by an agreement which provided that he should be liable for damage to the car unless he could show that the damage was not due to his act or default. While driving at night in an empty street 36 feet wide, the defendant swerved to avoid a cat which ran out suddenly into the road. Before he could apply his brakes, he hit a tree and damaged the car. Judge Gordon Clark held that the defendant was not at fault in swerving to avoid the cat and that he was not liable for the damage caused. Court: (CC) County Court Reported: (1958) 108 L.J. 348