In "First Blood", Rambo doesn't kill anyone and the movie climaxes with him crying and then getting arrested.
I bought a Rambo Blu-ray set last summer because I'm an 80s nostalgic headcase. I've only revisited the first one so far and, man, is it entertaining! And it's not a "white guys are awesome" movie. It's about someone who's totally screwed up in the head from Vietnam and it's sad. It's also a "F!ck the police" movie.
Rambo maims a lot of cops in it, but never kills anyone.