Can one give an immersion of exotic sphere $S^7$ in a standard sphere $S^8$ of radius $1$?

Yes. By SmaleHirsch theory it is enough to find a bundle injection $T\Sigma \to \epsilon^8$, so it is enough to find a trivialisation of $T\Sigma \oplus \epsilon^1$. It is a theorem of Kervaire and Milnor that every exotic sphere is stably framable, so $T\Sigma \oplus \epsilon^N$ is trivial for some large N, and the connectivity of $BO(8) \to BO$ means you can destabilise this to trivialise $T\Sigma \oplus \epsilon^1$. 


Slightly different: We computed the group of immersions of homotopy 4k1 spheres into $R^{4k+1}$ and also to some other euclidean spaces here: https://www.researchgate.net/publication/243028484_The_group_of_immersions_of_homotopy_4k1spheres 

