What does the Bhagavad Gita say about the soul being immortal?

In chapter 2, before any of the Gita's teaching on action or devotion begins, Krishna answers Arjuna's grief with a claim about what a person actually is. Arjuna has collapsed at the thought of killing his own kinsmen; Krishna's first move isn't tactical, it's metaphysical: na jayate mriyate va kadachin — “the soul is neither born, nor does it ever die; nor, having once existed, does it ever cease to be” (2.20). It is unborn (aja), eternal (nitya), everlasting (shashvata), and primeval (purana) — it is not slain when the body is slain. Krishna reinforces the point two verses later with a simile that became one of the Gita's most quoted images: just as a person casts off worn-out clothes and puts on new ones, the embodied soul casts off worn-out bodies and enters others (2.22). The argument isn't an excuse to treat violence lightly — the surrounding verses (2.19, 2.23–25) go on to say the soul cannot be cut by weapons, burned by fire, wetted by water, or dried by wind, precisely because it is not made of matter at all. What can be destroyed is the body; what grieves and fears is the identification with the body. Krishna's point to Arjuna is that clarity about this distinction — knowing which part of you is temporary and which is not — is the actual antidote to the grief and fear paralyzing him, not stoic suppression of feeling. This teaching sets up everything that follows in the Gita: karma yoga's instruction to act without anxious attachment to outcomes only makes sense once the deeper claim, that the true self was never at stake to begin with, has been established.