  We develop a theory of $p$-adic automorphic forms on unitary groups that
allows $p$-adic interpolation in families and holds for all primes $p$ that do
not ramify in the reflex field $E$ of the associated unitary Shimura variety.
If the ordinary locus is nonempty (a condition only met if $p$ splits
completely in $E$), we recover Hida's theory of $p$-adic automorphic forms,
which is defined over the ordinary locus. More generally, we work over the
$\mu$-ordinary locus, which is open and dense.
  By eliminating the splitting condition on $p$, our framework should allow
many results employing Hida's theory to extend to infinitely many more primes.
We also provide a construction of $p$-adic families of automorphic forms that
uses differential operators constructed in the paper. Our approach is to adapt
the methods of Hida and Katz to the more general $\mu$-ordinary setting, while
also building on papers of each author. Along the way, we encounter some
unexpected challenges and subtleties that do not arise in the ordinary setting.
