The mainstream consensus is that the Chinese word for lion was earliest attested as 「師子」 (Zhengzhang OC: /*sri ʔslɯʔ/) in literature roughly contemporary to the Book of Hàn:
Book of Han: Chronicles of the Western Regions
The lands of 烏弋* are hot in the summer, physically flat, and flourishing with vegetation. Its greenery, livestock, essential crops, fruits and vegetables, cuisine, palaces, marketplaces, currency, weapons, and treasures are the same as those of 罽賓†, and has [creatures such as] 桃拔‡, lions, and rhinoceros. The common folk enjoy blood sports.§ The obverse of their coinage has the likeness of a person's head, and their reverse depicts a horse-rider. Their weapons are adorned with gold and silver. [These lands] are extremely remote, and Hàn envoys rarely travel here; from the Yùmén or Yáng passes, through the Shànshàn Kingdom, to 烏弋山離 (Alexandria), one reaches the end of the southern path.
* /*qaː lɯɡ/, approximate English rendition: Alek, Alex. Abbreviation of 烏弋山離 (/*qaː lɯɡ sreːn rel/, Alexandria).
† /*krads mpin/, maybe Kashmir, Kabul, or Capisa
‡ Qílín-like creature
§ My own very liberal interpretation - may not be accurate
Alexandria points to one of the many settlements in current-day Afghanistan, at that time under the Parthian Empire, and the Parthian language is a Middle Western Iranian language, in the same family as modern Persian.
Based on all this, there is an idea that /*sri ʔslɯʔ/ is a loanword from Late Middle Persian /šēr/.