Yes, this is known. Raynaud showed that $B_{c_0}$ does not uniformly (in particular, bilipschitz) embed into any stable Banach space. $\ell_1$ is stable.

 - [Yves Raynaud, Espaces de Banach superstables, distances stables et homeomorphismes uniformes][1]
 


  [1]: https://link.springer.com/article/10.1007/BF02763170