  We give optimal bounds for the radial, space and time derivatives of
arbitrary order of the heat kernel of the Laplace--Beltrami operator on
Damek--Ricci spaces. In the case of symmetric spaces of rank one, these
complete and actually improve conjectured estimates by Anker and Ji. We also
provide asymptotics at infinity of all the radial and time derivates of the
kernel. Along the way, we provide sharp bounds for all the derivatives of the
Riemannian distance and obtain analogous bounds for those of the heat kernel of
the distinguished Laplacian.
