I know that we can have a $U(1)$ bundle over the base manifold like $S^2$. For example, we can choose the fibre $U(1)=S^1$ over $S^2$ as the $S^3$ Hopf fibration.
I know that we can have a $SU(2)$ bundle over the base manifold like $S^4$. For example, we can choose the fibre $SU(2)=S^3$ over $S^4$ as the $S^7$ Hopf fibration. Correct me if I was wrong!
Now, can we have any nontrivial $SU(2)$ bundles over a flat Euclidean $\mathbb{R}^d$ or Minkowski $\mathbb{R}^{d,1}$ space??
If so, could you give some examples?
If not, could you give me a proof? (or at least sketch your arguments to convince me?)