  Inelastic light scattering studies on single crystals of
(Na$_{1-x}$Li$_x$)$_2$IrO$_3$ ($x = 0, 0.05$ and $0.15$) show a polarization
independent broad band at $\sim $~2750 cm$^{-1}$ with a large band-width $\sim
1800$~cm$^{-1}$. For Na$_2$IrO$_3$ the broad band is seen for temperatures $
\leq 200$~K and persists inside the magnetically ordered state. For Li doped
samples, the intensity of this mode increases, shifts to lower wave-numbers and
persists to higher temperatures. Such a mode has recently been predicted
(Knolle et.al.) as a signature of the Kitaev spin liquid. We assign the
observation of the broad band to be a signature of strong Kitaev-exchange
correlations. The fact that the broad band persists even inside the
magnetically ordered state suggests that dynamically fluctuating moments
survive even below $T_{N}$. This is further supported by our mean field
calculations. The Raman response calculated in mean field theory shows that the
broad band predicted for the spin liquid state survives in the magnetically
ordered state near the zigzag-spin liquid phase boundary. A comparison with the
theoretical model gives an estimate of the Kitaev exchange interaction
parameter to be $J_K\approx 57$~meV.
