In a few anime and manga the protagonist or entire group are stuck in certain time event can't make time move forward. I mean whatever the protagonist does, the day is always September 2 all the time. In order to change that, he needs to do something. For example (Kurugaya route in Little Buster Viusal Novel, first volume of Hakomari Light Novel).
What were the first manga and anime produced with such trait?
According to an article in Japanese Wikipedia, critics like Hiroki Azuma and Masachi Osawa have cited Urusei Yatsura 2: Beautiful Dreamer (1984) as the "pioneering work" or "classical work" featuring this structure in the Japanese anime/manga field. After this movie, the otaku industry experienced a surge in the use of this plot device.
Arguably, the Chapter of Strange Beings (1981) in Osamu Tezuka's masterpiece Phoenix has a similar structure, but in this case:
the protagonist herself does not experience the loop, but is endlessly replaced by younger incarnation of herself.