We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a “typical” steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will never enter a vicinity of the original steady one. More precisely, we establish that there exist stationary solutions $$u_0$$ of the Euler equation on $${\mathbb {S}}^3$$ and divergence-free vector fields $$v_0$$ arbitrarily close to $$u_0$$ , whose (non-steady) evolution by the Euler flow cannot converge in the $$C^k$$ Hölder norm ( $$k>10$$ non-integer) to any stationary state in a small (but fixed a priori) $$C^k$$ -neighbourhood of $$u_0$$ . The set of such initial conditions $$v_0$$ is open and dense in the vicinity of $$u_0$$ . A similar (but weaker) statement also holds for the Euler flow on $${\mathbb {T}}^3$$ . Two essential ingredients in the proof of this result are a geometric description of all steady states near certain nondegenerate stationary solutions, and a KAM-type argument to generate knotted invariant tori from elliptic orbits.