1. Question Restatement:
Are there any non-biblical occurrences of Koine Greek? Are there occurrences prior to the Septuagint/NT?
2. Defining "Koine Greek":
I imagine the issue behind the question is the "artificial" use of the phrase "Koine Greek" in a way that excludes secular texts. That is because secular scholarship also uses different terms.
Koine Greek is also known as "Hellenistic Greek":
Koine Greek, Wikipedia - (UK English /ˈkɔɪniː/,2 US English /kɔɪˈneɪ/, /ˈkɔɪneɪ/ or /kiːˈniː/;2 from Koine Greek ἡ κοινὴ διάλεκτος, "the common dialect"), also known as Alexandrian dialect, common Attic, Hellenistic or Biblical Greek (Modern Greek: Ελληνιστική Κοινή, "Hellenistic Koiné", in the sense of "Hellenistic supraregional language"), was the common supra-regional form of Greek spoken and written during Hellenistic and Roman antiquity and the early Byzantine era, or Late Antiquity.
Referencing Non-Biblical Koine Texts is Very Necessary, to properly interpret New Testament texts:
The clear objection to the "theological use" of "Koine Greek" is that it has allowed "Christian" and "Jewish" specific grammars justify special meanings in religious texts only - to support certain doctrines. This allows theologians to claim "unique use / hapax" in Greek - when those phrases are not. (The most egregious issue is redefinitions of very common Greek words as "theological/spiritual", like "Xaris, Pistos, Dikaios, etc (Grace [instead of "Favor"], "Faith" [instead of Trust], "Righteous" [instead of "Just"])". These Greek words are redefined improperly used as purely theological concepts - to justify very specific theologies. These words, and many others, were not remotely used/interpreted with the theological connotations that were injected into them after.) We have several answers in this community that have wrongfully claimed "unique use / hapax" - because they do not consider these other texts.
The entire John 1:1 debate is ludicrous "word was a/g(G)od" - because of a theological refusal to appeal to Greek literature, or even other Semitic texts.
3. Answer, Other "Koine Greek" Texts:
From Wikipedia, Texts in Koine Greek:
Apocalypse of Peter
Discourses of Epictetus
Enchiridion of Epictetus
Exhortation to the Greeks
On the Sublime
Periplus of the Euxine Sea
Searching for texts using other names of "Koine Greek", (like Alexandrian Greek or Hellenistic Greek), will help identify these other secular texts).