Summary: We consider zero entropy \(C^{\infty}\)-diffeomorphisms on compact connected \(C^\infty\)-manifolds. We introduce the notion of polynomial growth of the derivative for such diffeomorphisms, and study it for diffeomorphisms which additionally preserve a smooth measure. We show that if a manifold~\(M\) admits an ergodic diffeomorphism with polynomial growth of the derivative then there exists a smooth flow with no fixed point on~\(M\). Moreover, if \(\dim M=2\), then it holds necessarily \(M={\mathbb T}^2\) and the diffeomorphism is \(C^{\infty}\)-conjugate to a skew product on the \(2\)-torus.