"Hypercontinuity" is a cardinality of a continuous set's power set (set of all subsets). When talking about fields, I mean the cardinality of field's set. At first glance there is nothing preventing them from existing, however I have never seen an example of one, a non-constructive proof that they exist, or a proof that they don't exist. Either of these would be appreciated.
That is basically all there is to the question itself, and the only thing I can add is my attempts to answer it myself.
- An attempt to construct one. Assume any continuous set $A$. We can construct a structure $\langle 2^A, \Delta, \cap \rangle$, where $2^A$ is a power set of $A$, which is hypercontinuous by the Cantor's theorem, $\Delta$ is symmetric difference of sets, which acts as addition of elements, and $\cap$ is intersection, which acts as multiplication. In this structure $\varnothing$ is an additive identity, and $A$ is a multiplicative identity. In such structure all of the field axioms are satisfied, except for the axiom of multiplicative inverses existance, making it a commutative ring instead of a field. In addition by assuming $A$ being a set of any cardinality, we can prove that commutative ring of any cardinality, representable as cardinality of some other set's power set, exists. Nothing about fields, however.
- Frobenius theorem. It states that under certain conditions any division ring is isomorphic either to $\Bbb R$, or to $\Bbb C$, or to $\Bbb H$. However one of the theorem conditions is that division ring has to be a finite-dimensional vector space over $\Bbb R$. And any such vector space is continuous, so this theorem is also not relevant to the question.
- Non-standard analysis. Unfortunately I am completely unfamiliar with this area of mathematics, however while researching on the question I stumbled upon a statement that gave me a vague idea that non-standard analysis might have an answer. I don't know if any of the structures it studies are fields, or if any are hypercontinuous, but I still thought it might be useful to mention it.