I bought my Corolla last year. I researched, online, various sites before I went. I got a price from edmunds.com, maybe, that had prices that others were paying in my area. I printed it out and took it with me.
The first dealership I went to, in my town, came close to that price. I tried negotiating and was told, by the salesman, that "look outside...does that sign say "Goodwill or Toyota"! I was shocked, to say the least! I got up, started walking out, and the manager stopped me. I guess the steam coming from my ears was a sign that I was pissed! I preceded to tell him, in front of all the other customers, that he should have a talk with his salespeople on what not to say to their customers. He tried to settle me down and make a deal, but, at that point, unless they were going to give me the car, I wasn't buying it there.
I went to another dealership from there. They were a little higher than the price I got off the website. I told him the price I got, they honored it, and I bought the car.
So, yes, you can negotiate a Corolla.
You don't know how much I wanted to drive to the first dealership, in my new Corolla, and say..."Remember me? I'm the Goodwill Girl...Big mistake...big mistake"! But I didn't.
Good luck!