A library has not been proved.
Debated. Burnt bronze sheets, and a bronze text found earlier in the same hall, make diggers think records were stored there.
The temple is at Kleidi-Samikon, in Ilia, Greece. It dates to the 6th century BC. It is about 28 metres long and 9.5 metres wide, with two large halls. A fire in the first half of the 3rd century BC brought the roof down and sealed what was on the floor. With the sheets were a bronze mirror, coins, and a stone seal. The page says this is believed to be the temple of Poseidon named by the old writer Strabo.
Whether that hall was truly an archive is still a suggestion.
https://www.ancient-origins.net/news-history-archaeology/samikon-temple-archive-00102592