I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns of $\nabla_p f(\bar p, \bar x)$ are in the column space of A. Couldn't we conclude that
$$ \nabla s(\bar p) = A^\dagger \nabla_p f(\bar p, \bar x), $$ where $\dagger$ represents the Moore-Penrose pseudo inverse.
If true, is there a good reference I can read to learn about such a result and its extensions?