I think so!
Cool question! This answer assumes water and carbon based life; I'm not an inorganic chemist and won't speak to other theories, but I'm sure some plausible ones exist!
An Earth-like planet
As other answers noted, one difficulty here is the pressure. According to this article, life has been shown to survive up to ~1 GPa for an extended period of time. But this is just Earth life which hasn't had much of a reason to adapt to extreme pressures. In the earth's mantle, the pressure ranges from 24 to 136 GPa. I don't think it's on face unreasonable to posit life that can adapt to these conditions; this paper showed that hydrocarbons are relatively stable under deep earth conditions.
The temperatures at these depths would be immense, from 500 - 4,200 K. However, as you can see from this phase diagram, there might be a sweet spot where at deep earth conditions water could exist as a supercritical fluid.
Maybe some kind of metabolism could develop under these conditions, but the massive amount of kinetic energy the molecules have would lead to very different reactions and thermodynamics from what we experience.
Smaller and lighter "planets"
Another thing to note is that most planets consist largely of heavier elements like metals and silicon. A planet made mostly of water and proteins would have much less gravity, and therefore pressure and temperature in its mantle, than a normal planet of a corresponding size. Finally, a living "planet" could be pretty huge, but still not exactly planet-sized. Planets, by definition, have to be big enough for their gravity make themselves roughly spherical and to clear their orbit. A living planet wouldn't have to worry about gravity doing these things, as its complicated biological processes could take care of them instead! A death star sized organism, for example, would hardly experience enough pressure to prevent biological activity, even at its core.
Of course, as others have mentioned, the living tissue could just be a shell, equivalent to the Earth's crust. I don't think this means that the planet isn't an organism; plenty of organisms have large quantities of nonliving structural material in them. You wouldn't say coral isn't an organism. In fact, the interior part of the planet might be secreted by the planet itself, and structured in a particular way to interact with the planet's core.
Such a planet might indeed lack a magnetosphere, which you mentioned would be deadly. There are a few options here:
- The planet has adapted to extreme radiation; perhaps all of its DNA is located way on the interior where the rays can't hit, or it's coated in some kind of hard carapace, or it just has extremely efficient DNA repair machinery. Radiation in space is only deadly to us because we don't put anymore energy than we need to into fighting it. In fact, maybe the planet uses this radiation to produce energy.
- The planet generates a magnetosphere without a core. This one is a little more far fetched. Instead of a huge molten spinning core producing energy, maybe the planet maintains a gigantic magnetic dipole of some kind in a cool core, which it spins around extremely quickly. This might make sense for a smaller planet.
Planet-Scale Metabolism
Two ways a planet-sized organism could generate energy for itself are by using the heat energy in its core or the energy of its sun. One idea for using the geothermal energy is a large ocean of water between the living tissue and the mantle/core. The huge convection currents of the water could interface with some kind of biological generator to produce energy; they could spin huge rotors or just move "organs" past each other at high speeds. As the cellular level, kinetic energy can be converted to chemical energy in a variety of ways. See ATP synthase, which is basically a wheel that creates ATP, the cell's energy currency, just by spinning. You could have a kind of reverse actin-myosin contraction, where two fibers of proteins smush ATP together when they are forced past each other by the current. Or the convection current could push water through symporters, using the kinetic energy of the water to push protons or what-have-you the other way, setting up a chemical gradient like the one used to harvest energy in all Earth life. As for the sun's light, the planet could just use good old photosynthesis.