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We report an in-plane thermal transport study of the honeycomb Kitaev material Na$_2$Co$_2$TeO$_6$ at subKelvin temperatures. In zero field, the $\kappa(T)$ displays a rather weak $T$-dependence but has a non-zero residual term $\kappa_0/T$, indicating strong phonon scattering by magnetic excitation and the possibility of itinerant spinon-like excitations coexisting with an antiferromagnetic order below 27 K. We propose the zero-field ground state is a novel fractionalized antiferromagnetic (AF*) state with both magnetic order and fractionalized excitations. With both the heat current and external field along the $a*$ (Co-Co bond) direction, the $\kappa_{a*}$ exhibits two sharp minima at 7.5 T and 10 T, and its value at 8.5 T is almost the same as the pure phononic transport for the high-field polarized state. This confirms the phase boundaries of the reported field-induced intermediate state and suggest its gapless continuum excitations possibly transport heat. No such intermediate phase was found in the $\kappa_a$ for the current and field along the $a$ (zigzag chain) direction. Finally, Na$_2$Co$_2$TeO$_6$ displays a strongly anisotropic magneto-thermal conductivity since the in-plane (out-of-plane) field strongly enhances (suppresses) the $\kappa_{a*}$ and $\kappa_a$., Comment: 7 pages, 3 figures