  There are examples of complete spacelike surfaces in the Lorentzian product
$\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K\leq -1$.
In this paper, we show that there exists no complete spacelike surface in
$\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K>-1$.
