Is $2$ not a prime (in $\Bbb R$)?
Considering the definitions of irreducible and prime elements in an integral domain $R$ and knowing the fact
Every Field is an integral domain
Does it mean 2 is not a prime in $\mathbb{R}$?
Since, for 2 to be a prime element in $\mathbb{R}$, it has to be a non-unit element. But, 2 is a unit.
Ring theory is very boring in fields, since, as you mention, every non-zero element is a unit. It does follow as you say that there are no primes in a field, simply because there are no non-units other than zero.