The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category [formula omitted] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that [formula omitted] can be endowed with a Chow weight structure w_Chow whose heart is Chow^eff[1/p] (weight structures were introduced in a preceding paper, where the existence of w_Chow for [formula omitted] was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor [formula omitted]→K^b (Chow^eff [1/p]) (which induces an isomorphism on K₀-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on [formula omitted]. We also mention a certain Chow t-structure for [formula omitted] and relate it with unramified cohomology.