Let ${\cal U}$ be a monster model. Let $p(x;z)\subseteq L$ be a type. Is the following true?
If the set $\big\{p({\cal U}^{|x|};a)\ :\ a\in{\cal U}^{|z|}\big\}$ is infinite, then it has cardinality $|{\cal U}|$.
The question makes sense also when $x$ and $z$ are infinite tuples (then I conjecture the answer is `no').
I'm most interested in the case $|x|,|z|<\omega$.
No, not even for finite tuples. Consider the language $\{E_n\mid n\in \omega\}$ and the theory asserting that each $E_n$ is an equivalence relation, $E_0$ has just one class, and $E_{n+1}$ refines each $E_n$-class into two classes, for all $n$.
Let $p(x,y) = \{E_n(x,y)\mid n\in \omega\}$. So $p$ defines the equivalence relation $\bigcap_{n\in\omega} E_n$. This is probably the simplest example of a type-definable equivalence relation which is not definable.
Even in the monster model, $p(x,y)$ has only $2^{\aleph_0}$ equivalence classes, corresponding to the paths through a complete binary tree, with the nodes at level $n$ corresponding to the $E_n$-classes. So $$|\{p(\mathcal{U};a) \mid a\in \mathcal{U}\}| = 2^{\aleph_0}.$$