I've read the following proof-less claim: there is no operad such that the algebras over it are fields. We can make that precise by asking whether there's an operad $\mathcal{P}$ in abelian groups such that the category $\mathcal{P}Alg$ is equivalent to the category of fields.
Intuitively this seems true: every axiom but the existence of inverses would be fine (and indeed there's an operad for commutative rings). But this isn't a proof, as there could be a presentation of field axioms that would translate to an operad.