To study diffusion processes on the p-Wasserstein space $\mathscr P_p$ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathscr P_p,Λ)$ for a reference probability $Λ$ on $\mathscr P_p$. It is formulated in terms of an upper bound condition with the uniform norm of the intrinsic derivative. We find a versatile class of quasi-regular local Dirichlet forms on $\mathscr P_p$ by using images of Dirichlet forms on the tangent space $L^p(X\to X,μ_0)$ at a reference point $μ_0\in\mathscr P_p$. The Ornstein--Uhlenbeck type Dirichlet form and process on $\mathscr P_2$ are an important example in this class. We derive an $L^2$-estimate for the corresponding heat kernel and an integration by parts formula for the invariant measure.