Oh, yes. The movie that you are looking for is "The Bad Seed" based on the stage play by Maxwell Anderson. There have been two versions. The 1956 theatrical version with Nancy Kelly as the girl's mother, Patty McCormack as the girl, Henry Jones as the man who saw too much, Eileen Heckhart as the dead kid's mother, and William Hopper as the girl's father. The other version is a 1985 TV movie with Blair Brown as the girl's mother, Christa Danton as the girl. and David Carradine as the man who saw too much. (IMHO) the 1956 version is the superior of the two versions. There was also a 1963 Turkish film version which I have not seen. As an added note, I think the 1956 version is one of, if not the first time, that a child has been portrayed as a multiple murderer.
You are indeed correct. No film of it's type existed at that time, and it was a shocker to be reckoned with. NO film of it's day portrayed a child as a killer. Despite the years that have gone by, the film's impact is still very powerful. Some films make that lasting impression and this is one of them.