Critics who claim that references to "elephants" in the Book of Mormon are anachronisms ignore important contradictory evidences.
Elephants are only mentioned once in the Book of Mormon in connection with the Jaredites who arrived in the New World between 2600 and 2100 BC. Within the Jaredite record (the Book of Ether), they are not prominent.
There is no mention of elephants during the later Nephite period. It appears that within a few centuries they became extinct. While there may have been large numbers in the Jaredite period, their extinction would support that overall their numbers were fewer making it more difficult to find archaeological evidence of their existence millennia later.
We don’t know where the Jaredites lived so it is impossible to excavate specifically to look for the bones of elephants. An infinitesimally small portion of possible geographical locations where the Jaredites may have lived have been explored by archaeologists.
The bones of elephantine mammals have been found in Ecuador and elsewhere in South America some dating to 3700 years ago.
Evidence for the survival of the elephant can be found in Native American myths and traditions. Oral traditions have been documented for Native American groups from Canada to the Gulf of Mexico persuading some scholars that they are based upon a core memory of actual historical encounters with elephant-like species who may have survived into the region as late as 3,000 years ago.
The Book of Mormon describes animals as existing several thousand years ago. It is impossible to prove they did not exist. The process becomes more difficult if the translation of languages is involved because it may be impossible to accurately identify the thing reportedly absent.
