  We formulate a free probabilistic analog of the Wasserstein manifold on
$\mathbb{R}^d$ (the formal Riemannian manifold of smooth probability densities
on $\mathbb{R}^d$), and we use it to study smooth non-commutative transport of
measure. The points of the free Wasserstein manifold
$\mathscr{W}(\mathbb{R}^{*d})$ are smooth tracial non-commutative functions $V$
with quadratic growth at $\infty$, which correspond to minus the log-density in
the classical setting. The space of smooth tracial non-commutative functions
used here is a new one whose definition and basic properties we develop in the
paper; they are scalar-valued functions of self-adjoint $d$-tuples from
arbitrary tracial von Neumann algebras that can be approximated by trace
polynomials. The space of non-commutative diffeomorphisms
$\mathscr{D}(\mathbb{R}^{*d})$ acts on $\mathscr{W}(\mathbb{R}^{*d})$ by
transport, and the basic relationship between tangent vectors for
$\mathscr{D}(\mathbb{R}^{*d})$ and tangent vectors for
$\mathscr{W}(\mathbb{R}^{*d})$ is described using the Laplacian $L_V$
associated to $V$ and its pseudo-inverse $\Psi_V$ (when defined). Following
similar arguments to arXiv:1204.2182, arXiv:1701.00132, and arXiv:1906.10051 in
the new setting, we give a rigorous proof for the existence of smooth transport
along any path $t \mapsto V_t$ when $V$ is sufficiently close $(1/2) \sum_j
\operatorname{tr}(x_j^2)$, as well as smooth triangular transport.
