Theory with a single elementarily minimal model and no prime model I'm looking for a Theory in 1st Order Logic with no prime models and precisely one elementarily minimal model.
The examples with elementarily min but no prime models which spring to mind are Theory of fields/ Theories of Algebraically Closed Fields (both have countably many Elem. Min. Models), or messing about with constant symbols in our language (I've seen an example with  2 Elem Min models, no prime, but this doesn't appear to be generalizable down to one Elem min model)
 A: If you allow incomplete theories (and it seems you do, since you consider the theory of fields in the question), then it's easy to come up with such an example. The point is that an incomplete theory can never have a prime model, so all we have to do is ensure that $T$ is incomplete and only one of its completions has an elementarily minimal model.
Let $L$ be the language with constant symbols $(c_n)_{n\in \mathbb{N}}$, and consider the $L$-theory $T$ axiomatized by: $$\exists x_0\dots x_{n-1} \bigwedge_{i<j<n} x_i\neq x_j\\c_0 = c_1 \rightarrow (c_i = c_j) \quad \text{for all }i\neq j\\
c_0\neq c_1\rightarrow (c_i\neq c_j)\quad \text{for all }i\neq j$$
A model of $T$ is an infinite set with specified elements $(c_n)_{n\in \mathbb{N}}$ such that either all the $c_i$ are equal, or all the $c_i$ are distinct. The unique elementarily minimal model $M$ of $T$ is the one where every element of $M$ is named by a constant symbol. And $T$ is incomplete, so it has no prime model.
A more difficult challenge would be to come up with an example of a complete theory satisfying your criterion.
