  It is known that the gauge field and its composite operators evolved by the
Yang--Mills gradient flow are ultraviolet (UV) finite without any
multiplicative wave function renormalization. In this paper, we prove that the
gradient flow in the 2D $O(N)$ non-linear sigma model possesses a similar
property: The flowed $N$-vector field and its composite operators are UV finite
without multiplicative wave function renormalization. Our proof in all orders
of perturbation theory uses a $(2+1)$-dimensional field theoretical
representation of the gradient flow, which possesses local gauge invariance
without gauge field. As application of the UV finiteness of the gradient flow,
we construct the energy--momentum tensor in the lattice formulation of the
$O(N)$ non-linear sigma model that automatically restores the correct
normalization and the conservation law in the continuum limit.
