  Suppose that T_t is a symmetric diffusion semigroup on L^2(X) and consider
its tensor product extension to the Bochner space L^p(X,B), where B belongs to
a certain broad class of UMD spaces. We prove a vector-valued version of the
Hopf--Dunford--Schwartz ergodic theorem and show that this extends to a maximal
theorem for analytic continuations of the semigroup's extension to L^p(X,B). As
an application, we show that such continuations exhibit pointwise convergence.
