  We consider Shimura varieties associated to a unitary group of signature
$(n-1, 1)$. For these varieties, we construct $p$-adic integral models over odd
primes $p$ which ramify in the imaginary quadratic field with level subgroup at
$p$ given by the stabilizer of a vertex lattice in the hermitian space. Our
models are given by a variation of the construction of the splitting models of
Pappas-Rapoport and they have a simple moduli theoretic description. By an
explicit calculation, we show that these splitting models are normal, flat,
Cohen-Macaulay and with reduced special fiber. In fact, they have relatively
simple singularities: we show that a single blow-up along a smooth codimension
one subvariety of the special fiber produces a semi-stable model. This also
implies the existence of semi-stable models of the corresponding Shimura
varieties.
