Let a Lipschitz polygon or polyhedron $\Omega$, a function ${\boldsymbol v}$ in the Sobolev space ${\boldsymbol H}(\mathrm{div},\Omega)$, and a simplicial mesh of $\Omega$ be given. We prove the equivalence of two piecewise (Raviart--Thomas) polynomial best approximations of ${\boldsymbol v}$: 1) globally on the whole computational domain $\Omega$, with the normal trace continuity requirement and a divergence constraint; 2) locally on each mesh element, without any interelement continuity requirement and without any constraint on the divergence. The former (global-best continuous constrained piecewise polynomial approximation) arises in numerical methods for partial differential equations related to the ${\boldsymbol H}(\mathrm{div},\Omega)$ space, whereas the latter (local-best discontinuous unconstrained piecewise polynomial approximation) is a key quantity in approximation theory. Crucially, we establish $p$-robustness in that the equivalence constant only depends on the mesh shape regularity and the spatial dimension. This improves the recent result of [IMA J. Numer. Anal. {\bf 42} (2022), 1023--1049], where the equivalence constant was possibly dependent on the underlying polynomial degree. Consequently, we obtain fully $h$- and $p$- (mesh-size- and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only requested separately on each mesh element. These two results immediately follow by our construction of an operator from the infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{div},\Omega)$ to its finite-dimensional Raviart--Thomas subspace that has the following properties: 1) it is defined over the entire ${\boldsymbol H}(\mathrm{div},\Omega)$ and preserves boundary conditions imposed on a part of the boundary of $\Omega$; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on elementwise ${\boldsymbol L}^2$-orthogonal polynomial projections; 4) it is a projector, i.e., it leaves intact objects that are already in the Raviart--Thomas piecewise polynomial space; 5) it is locally and $p$-robustly stable in the ${\boldsymbol L}^2$-norm, up to $hp$ data oscillation; 6) its approximation property is locally and $p$-robustly equivalent to that of the discontinuous unconstrained (elementwise ${\boldsymbol L}^2$-orthogonal) projection; 7) it satisfies the commuting property with the $L^2$-orthogonal projection onto piecewise polynomials.