  We prove a quantitative version of a Gidas-Ni-Nirenberg-type symmetry result
involving the $p$-Laplacian.
  Quantitative stability is achieved here via integral identities based on the
proof of rigidity established by J. Serra in 2013, which extended to general
dimension and the $p$-Laplacian operator an argument proposed by P. L. Lions in
dimension $2$ for the classical Laplacian.
  Stability results for the classical Gidas-Ni-Nirenberg symmetry theorem
(involving the classical Laplacian) via the method of moving planes were
established by Rosset in 1994 and by Ciraolo, Cozzi, Perugini, Pollastro in
2024.
  To the authors' knowledge, the present paper provides the first quantitative
Gidas-Ni-Nirenberg-type result involving the $p$-Laplacian for $p \neq 2$. Even
for the classical Laplacian (i.e., for $p=2$), this is the first time that
integral identities are used to achieve stability for a Gidas-Ni-Nirenberg-type
result.
  In passing, we obtain a quantitative estimate for the measure of the singular
set and an explicit uniform gradient bound.
