The effect of off-plane impurity on superconductivity and non-Fermi-liquid (NFL) behavior in the layered heavy-fermion compound CeCo$_{1-x}$Ni$_x$In$_5$ is investigated by specific heat, magnetization, and electrical resistivity measurements. These measurements reveal that the superconducting (SC) transition temperature T$_c$ monotonically decreases from 2.3 K (x=0) to 0.8 K (x=0.20) with increasing x, and then the SC order disappears above x=0.25. At the same time, the Ni substitution yields the NFL behavior at zero field for x=0.25, characterized by the -ln T divergence in specific heat divided by temperature, C$_p$/T, and magnetic susceptibility, M/B. The NFL behavior in magnetic fields for x=0.25 is quite similar to that seen at around the SC upper critical field in pure CeCoIn$_5$, suggesting that both compounds are governed by the same antiferromagnetic quantum criticality. The resemblance of the doping effect on the SC order among Ni- , Sn-, and Pt-substituted CeCoIn5 supports the argument that the doped carriers are primarily responsible for the breakdown of the SC order. The present investigation further reveals the quantitative differences in the trends of the suppression of superconductivity between Ce(Co,Ni)In$_5$ and the other alloys, such as the rates of decrease in T$_c$, dT$_c$/dx, and specific heat jump at T$_c$, d($Delta$C$_p$/T$_c$)/dx. We suggest that the occupied positions of the doped ions play an important role in the origin of these differences.
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