  We have systematically studied the low-temperature specific heat of the
BaFe$_{2-x}$Ni$_x$As$_2$ single crystals covering the whole superconducting
dome. Using the nonsuperconducting heavily overdoped x = 0.3 sample as a
reference for the phonon contribution to the specific heat, we find that the
normal-state electronic specific heats in the superconducting samples may have
a nonlinear temperature dependence, which challenges previous results in the
electron-doped Ba-122 iron-based superconductors. A model based on the presence
of ferromagnetic spin fluctuations may explain the data between x = 0.1 and x =
0.15, suggesting the important role of Fermi-surface topology in understanding
the normal-state electronic states.
