Summary: We consider functions \(u \in L^{\infty}(L^{2}) \cap L^{p}(W^{1,p})\) with \(1<p< \infty\) on a time-space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation \(u_{\lambda}\) of \(u\) which coincides with \(u\) on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of L. Diening et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9, No. 1, 1–46 (2010; Zbl 1253.76017)]. Since div \(u_{\lambda}=0\), we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of D. Breit et al. [J. Differ. Equations 253, No. 6, 1910–1942 (2012; Zbl 1245.35080)].