Consider the theory of two equal-cardinality infinite sets:
* The language has two unary predicates $U$ and $V$ and a binary relation $E$.
* The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, _every_ model has infinite coinfinite definable sets!
* * *
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.