  The honeycomb-lattice cobaltate Na$_3$Co$_2$SbO$_6$ has recently been
proposed to be a proximate Kitaev quantum spin liquid~(QSL) candidate. However,
non-Kitaev terms in the Hamiltonian lead to a zigzag-type
antiferromagnetic~(AFM) order at low temperatures. Here, we partially
substitute magnetic Co$^{2+}$ with nonmagnetic Zn$^{2+}$ and investigate the
chemical doping effect in tuning the magnetic ground states of
Na$_3$Co$_{2-x}$Zn$_x$SbO$_6$. X-ray diffraction characterizations reveal no
structural transition but quite tiny changes on the lattice parameters over our
substitution range $0\leq x\leq0.4$. Magnetic susceptibility and specific heat
results both show that AFM transition temperature is continuously suppressed
with increasing Zn content $x$ and neither long-range magnetic order nor spin
freezing is observed when $x\geq0.2$. More importantly, a linear term of the
specific heat representing fermionic excitations is captured below 5~K in the
magnetically disordered regime, as opposed to the $C_{\rm m}\propto T^3$
behavior expected for bosonic excitations in the AFM state. Based on the data
above, we establish a magnetic phase diagram of Na$_3$Co$_{2-x}$Zn$_x$SbO$_6$.
Our results indicate the presence of gapless fractional excitations in the
samples with no magnetic order, evidencing a potential QSL state induced by
doping in a Kitaev system.
