We have systematically studied the low-temperature specific heat of the BaFe2−xNixAs2 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.