A Walsh zero space (WZ space) for f : F2n → F2n is an n-dimensional vector subspace of F2n x F2n whose all nonzero elements are Walsh zeros of f. We provide several theoretical and computer-free constructions of WZ spaces for Gold APN functions f(x)=x2i+1 on F2n where n is odd and gcd(i,n)=1. We also provide several constructions of trivially intersecting pairs of such spaces. We illustrate applications of our constructions that include constructing APN permutations that are CCZ equivalent to f but not extended affine equivalent to f or its compositional inverse.