I have hit many animals in my car but don't feel good about it. Where I live we have very narrow lanes enclosed by tall stone hedges and animals just dart out in front of your car and there very little you can do to avoid a collision. Once, on my way to work pre-dawn, I hit 21 rabbits. I wasn't exactly happy about it but there was nothing I could have done, apart from walk the 15 miles to work.
On another trip to the same place of work (and on a wider section of road) I saw a cat sat in my lane. I slowed down and swerved to the other side of the road and the bloody thing ran right across the road, under my wheels. I did my best to avoid it but had no chance.
Strangely, in the UK, you do not have to report that you have run over a cat.
"A dog (as well as a goat, horse, cattle, ass, mule, sheep and pig) does come within the remit of the Road Traffic Act and is a reportable accident so you are required by law to report it to the police. "
I've taken home a roadkill deer before. It kept me in venison stew for the winter.
Here's is a typical Cornish lane to illustrate my point:

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