  In this paper we show that certain Shimura varieties, uniformized by the
product of complex unit balls, can be p-adically uniformized by the product (of
equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic
uniformization to automorphic vector bundles. It is a continuation of our
previous work [V], and contains all cases (up to a central modification) of a
uniformization by known p-adic symmetric spaces. The idea of the proof is to
show that an arithmetic quotient of the product of Drinfeld upper half-spaces
cannot be anything else than a certain unitary Shimura variety. Moreover, we
show that difficult theorems of Yau and Kottwitz appearing in [V] may be
avoided.
