Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, $J\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$, where $r$ is a distance, we show that for spin anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}exp(\ensuremath{-}\ensuremath{\alpha}+2)$ the system exhibits a quasiballistic spin transport and the presence of fermionic excitation, which do not decay up to extremely long times $\ensuremath{\sim}{10}^{3}/J$. This conclusion is reached on the basis of the dynamics of spin domains, the dynamical spin conductivity, inspection of the matrix elements of the spin-current operator, and analysis of most conserved operators. Our results smoothly connect two models in which fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.