Abstract The most fascinating feature of certain two-dimensional (2D) gapless quantum spin liquid (QSL) is that their spinon excitations behave like the fermionic carriers of a paramagnetic metal. The spinon Fermi surface is then expected to produce a linear increase of the thermal conductivity with temperature that should manifest via a residual value ( κ 0 / T ) in the zero-temperature limit. However, this linear in T behavior has been reported for very few QSL candidates. Here, we studied the ultralow-temperature thermal conductivity of an effective spin-1/2 triangular QSL candidate Na 2 BaCo(PO 4 ) 2 , which has an antiferromagnetic order at very low temperature ( T N ~ 148 mK), and observed a finite κ 0 / T extrapolated from the data above T N . Moreover, while approaching zero temperature, it exhibits series of quantum spin state transitions with applied field along the c axis. These observations indicate that Na 2 BaCo(PO 4 ) 2 possibly behaves as a gapless QSL with itinerant spin excitations above T N and its strong quantum spin fluctuations persist below T N .