Murder at the Gallop from 1963 with Margaret Rutherford, Charles Tingwell and Stringer Davis
There were four Agatha Christie Miss Marple movies made in the early sixties; based on the two I've seen, they are fun-enough efforts if thought of as good B-movies or TV-shows that happened to be made in a movie format.
The general story seems to be the same in the two I've seen so far. Miss Marple, a grandmotherly looking woman who lives in a small English village, stumbles upon a murder and doubts that the police are conducting a thorough-enough investigation. So, she pursues her own inquiry getting underfoot of the police inspector, Charles Tingwell, who likes Miss Marple, but is often irritated by her, to be honest, constantly upstaging him.
It's all lighthearted stuff as the fun is seeing a scripted-out-of-central-casting grandmother - as a somewhat antecedent of the Columbo-TV model - seemingly harmlessly following clues and asking questions as she's all but dismissed by everyone until, right at the end, they realize that grandma has figured it all out.
In Murder at the Gallop, the story is set amongst the horsey set as a patriarch of a wealthy family is murdered with all his relatives, and a few other people, suspects. To tell more isn't so much to risk giving anything away, but wasting effort as you've seen some version of the plot fifty or more times if you've watched any TV-detective-mystery dramas from the sixties through the nineties. Here, eventually, usually after an attempt is made on Miss Marple's life by the murderer, she, in her understated and almost clumsy manner, gathers everyone together in one room and exposes the guilty person.
In these films, there's a delicate balance between tongue-in-cheek whimsy and mystery story that leans toward the former but nods enough at the latter to hold the two ends together. But you don't watch a Miss Marple film for the story, you watch it because you enjoy the eccentric Miss Marple - it either works for you or it doesn't. I can only enjoy them with plenty of time between each one.
But if their quirkiness does appeal to you, the chemistry between the frustrated young police inspector and septuagenarian Miss Marple is movie gold. Not since Marie Dressler in the 1930s has there been an elderly female star who could carry a movie with such presence and personality. Plus, for us today, the time travel these movies provide to a small village in early '60s England is pure fun.