Was watching Rick and Morty on the Season 3 premiere and they have that very small planet that the family goes to to escape. The planet is humorously small in that it is noticeable round while walking. The planet also apparently has animals and breathable air for humans. At one point, Rick goes to the South Pole of the planet and goes into a cave that takes him to the core of the planet which is shown as being smaller than him, from what I remember and what it looked like.
Could a planet like this actually exist with all of these features, only being a few acres in size, at most? Would a breathable atmosphere be possible? Would a core be present at all?
To put it in more realistic terms, the planet would be the size of Manhattan Island in New York City but folded to be round.
Accepting the surface gravity limit of 1.25 to 1.5 g and the minimum escape velocity of 6.25 kilometers per second from Habitable planets for Man Stephen H. Dole, 1964,
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
Let's use these calculators to find these variables
https://philip-p-ide.uk/doku.php/blog/articles/software/surface_gravity_calc
https://www.omnicalculator.com/physics/escape-velocity
I'm using 22km for the length of Manhattan which gives us a planetary radius of ~3.5km
3.5km makes it impossible to have both an acceptable surface gravity and escape velocity for an atmosphere. I was looking at a 3e-7 earth mass which provides the correct gravity but it's too low of an escape velocity. Raising the mass increases the gravity very quickly. Even so a 3e-7 earth mass is a density of 10e7 kg/m^3 which is in the red dwarf density. I am not skilled enough in astrophysics to be certain but it seems to object would undergo some sort gravitational collapse
Disclaimer: I quit astrophysics a long time back when I moved into mathematical physics (which I only recently quit)
If it's in red dwarf territory then I think it's too light to be stable and would just explode rather than collapse because that mass is far too small and the electrostatic force too strong compared to gravity. Don't take this as fact because I'm just estimating based on my intuition about the relative strengths of those forces, though I am reasonably confident.
This does bring up the fact that white dwarves will eventually (trillions of years iirc) cool down and basically be big inert lumps of dead star. Thinking about the cold iron cores of stars was what eventually led to the discovery of things like neutron stars and black holes.