“Can local anesthesia cause nerve damage?”

Wrong additive drugs, wrong site injection, needle itself can cause nerve damage.

Using a needle to inject a volume of fluid, if given directly into a nerve can cause nerve damage just by the trauma of the needle and the volume of the injectate, irrespective of whether local anesthetic is present or not. Local anesthetics themselves are not toxic to nerves. Most physicians are careful enough to use small needles, ultrasound guidance, preservative-free local anesthetic, in small volumes around the nerves in a manner that avoids all potential nerve damage.

Yes, if it is injected into the nerve

It is possible if the medication is injected into the nerve directly