According to this Reddit thread, the answer is no, vapor pressure can't be zero when temperature is above absolute zero. I suspect the answer might actually be yes according to a precise definition I will give later before I state what I'm really asking.
Because of the existence of gravity and the cosmological constant and because temperature might not be well defined because the zeroth law of thermodynamics hasn't been proven to be an absolute law according to this answer, we can't even define the vapour pressure of any substance sufficiently close to absolute zero because the bigger an object is, the closer to absolute zero it can get but any solid spherical object in its stable form with no pores in it of sufficient size will collapse into a black hole. I believe we fully understand a simplification of the theory of the universe that doesn't include gravity, the cosmological constant, or dark matter but I'm not sure of that because people seem to be using the Hadron collider to make new observations and learn more about the theory. However, I'm pretty sure that in that theory, we can define temperature in such a way that two substances in thermal equilibrium don't differ in Kelivn temperature by more than 8% and the intensity of blackbody radiation at any wavelength with more than 96% absorption is within 8% of what it's predicted to be.
I believe that theory almost always simulates an even simpler 3-dimensional quantum mechanical theory where electrons and nuclei are point charges and there's no nuclear chemistry. That 3-dimensional quantum mechanical theory can in turn be approximated by a simple nonrelativistic quantum mechanical theory because the fine structure constant is so small.
In nonrelativistic quantum mechanics, there are no photons and no blackbody radiation. Mercury is severely affected by the relativistic effects but it can probably be easily shown that none of the light elements are. However, for any specific substance, we can define the temperature of that specific substance in that simple theory to be the limit of what's it's defined to be in a simplified relativistic quantum mechanical theory when you set the speed of light close enough to infinite while keeping the proton mass; electron mass; proton charge; Coulumb's constant, and Planck's constant the same. For any substance, if you set the speed of light close enough to infinity, temperature is probably defined in such a way that substance's greybody radiation at any wavelength will be within 8% of prediction. According to that definition, the larger an object is, the closer to absolute zero it can get. Also according to this question, a surface of a given solid might even have a different vapour pressure depending on its orientation with respect to its crystal lattice in the simple theory where the vapour pressure of the surface whose orientation has the highest vapour pressure is at the highest pressure for its internal energy that gives a zero rate of homogeneous nucleation of the solid phase, as long as a shape whose faces are all at the orientation with the highest vapour pressure exists. I define the vapour pressure of any surface of a stable solid at any temperature above absolute zero to be the limit as its size approaches infinity of its vapour pressure at that temperature and orientation. My question is
According to the simple nonrelativistic theory, do you know how to invent a precise definition of temperature and prove entirely mathematically that the theory predicts that it has certain properties such as approximately following the zeroth law of thermodynamics on the Kelvin scale, and then prove or disprove entirely mathematically that that theory predicts that vapour pressure of any surface of a solid at any orientation above absolute zero must be nonzero without using the assumption that since something was observed in experiment, the theory predicts it?
I don't see why a substance must always have nonzero vapour pressure above absolute zero. If a solid has a microscopically thin layer of hexane on it, we could observe water to have a different contact angle with it than the theory predicts it to have. Also according to this answer, it hasn't been proven to be consistent with the simple nonrelativistic theory that a lot of the statements we were taught such as the second law of thermodynamics are all true. If you assume that a substance above absolute zero is in equilibrium and its equibrium state has certain chatoic properties where heat diffuses normally and its equilibrium state is time-invertable, then the substance must have nonzero vapour pressure. Since heat can diffuse, the reverse process can also occurr where a lot of heat randomly concentrates in a small area at the surface and knocks a surface atom loose.
I don't see why if it's in equilibrium, its equilibrium state must be time-invertable. Maybe there is a substance that at any sufficiently low temperature has a tendency to exponentially approach a non-time-invertible state where all the atoms are in phase at all times and repeat actions at regular time intervals and therefore has zero vapour pressure.