We have investigated the low temperature specific heat properties as a function of magnetic field in the strongly correlated layered cobalt oxide [BiBa$_{0.66}$K$_{0.36}$O$_2$]CoO$_2$. These measurements reveal two kinds of magnetic field dependent contributions in qualitative agreement with the presence of a previously inferred magnetic Quantum Critical Point (QCP). First, the coefficient of the low temperature T$^3$ behavior of the specific heat turns out to sizeably decrease near a magnetic field consistent with the critical value reported in a recent paper. In addition, a moderate but significant enhancement of the Sommerfeld coefficient is found in the vicinity of the QCP suggesting a slight increase of the electronic effective mass. This result contrasts with the divergent behavior of the previously reported Pauli susceptibility. Thus, a strongly enhanced Wilson ratio is deduced, suggesting efficient ferromagnetic fluctuations in the Fermi liquid regime which could explain the unusual magnetic field dependent specific heat. As a strong check, the high magnetic field Wilson ratio asymptotically recovers the universal limit of the local Fermi liquid against ferromagnetism.
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