Conceivably you could have life within molten lava. This life would be silicone based. It's a staple of science fiction of course.
But there's never been anything resembling a trace of a living creature found in lava to date. It's something briefly discussed, and dismissed as extremely unlikely, as a possibility for lava flows on Venus here: Cosmic Biology
As for the core, temperatures far higher. Possibly 6,000 C.
But - then - that means - realms that are very hard to explore and duplicate on the surface. At those high pressures, millions of times the surface pressure - then iron is solid in the core, even at this very high temperature. Which Material Has the Highest Melting Point?
Does that make any difference, can you have living creatures somehow using solids - and some kind of chemistry - that can exist at these enormously high temperatures and pressures?
I don't know the answer, haven't seen much written about it outside of science fiction.
One objection sometimes made to over simple speculations about life that might exist in regimes way outside of the limits of Earth life is that we have been able to explore a vast complex organic chemistry just because our labs are places where it is easy to explore chemistry at the temperatures and pressures that are ideal for organics.
So - is it partly an observer effect, that we know about the complex chemistry of organics just because our laboratories are ideal places for exploring it?
Might there be an equally complex chemistry at say temperatures and pressures of the Earth's core, in your question - or in lava with silicones, or at ultra low temperatures or whatever, liquid nitrogen, that we don't notice because those conditions are so hard to replicate in our laboratories so we do comparatively simple experiments in them?