That's pretty nervy to ask a co-worker to borrow a car.
I think you are very sensible. The only way Mike and I would loan our car---is if my in-law's needed it. They are the only family that live close, and since they have 3 cars it's pretty unlikely.
They were very generous a few years ago when we had a car that was constantly in the shop. So I would loan them my car in an emergency.
ETA: We are a 1 car family, so I'm more protective of our car than I think I would be if we had 2 cars.
I'm the same way. I dont like people driving my cars. Rob has no problem lending the car to a friend or someone, but I hate it. That's 2000lbs of steel you are lending to someone, and anything can happen if they arent paying attention.
I don't think that your being unreasonable, I wouldn't have let her borrow my car either if I were you. I think that asking to use someone's car (if its not an absolute emergency) is rude, I would never ask to borrow someone elses car. Borrowing expensive items like a car only leads to problems should something go wrong.