So... is Esper Mami finished or not?
NO, it is not finished.
Why?
The anime finished after the manga:
Anime finished: October 26, 1989
Manga finished: January 01, 1983
... But MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Esper Mami follows the adventures of Mami Sakura, a regular middle schooler who suddenly develops psychic abilities.
She partners with her friend, Takahata, to use these newfound powers to help people and solve strange mysteries in their town.
Whenever Mami senses trouble, she uses a special device to teleport herself to the scene.
With her telekinetic and telepathic skills, she works to protect those in need from various unusual situations.