If $\kappa$ is an infinite cardinal, we say a model $\mathcal M$ is $\kappa$-saturated when for all $A\subset M$ with $|A|<\kappa$ all complete $1$-types $p(x)$ in $\mathcal L_A$ are realised in $\mathcal M$.
Furthermore $\mathcal M$ is just called saturated when it is $|\mathcal M|$-saturated.
Is it also possible for a model to be $|\mathcal M|^+$-saturated?
On
No, because then we can take a type $$ p(x) = \{x \neq a : a \in \mathcal M\}. $$ This is finitely realisable in $\mathcal M$ (assuming $\mathcal M$ is infinite), but clearly not realisable in $\mathcal M$.
If you really want it to be a complete type, you can just use that any partial type can be extended to a complete one.
Consider the (partial) type $$p_\mathcal{M}=\{x\not=a: a\in\mathcal{M}\}.$$ Since $\mathcal{M}$ is infinite, $p$ is indeed finitely consistent over $\mathcal{M}$, but cannot be realized.
In fact, in some sense this is the canonical example of an "interesting" partial type: given a parameter set $A$, there are two kinds of type over $A$ - those which have a realization in $A$ and those which (regardless of whether they're realized in the ambient model) don't. Every type of the latter kind extends the analogous partial type $$p_A=\{x\not=a:a\in A\}.$$