Consistently, $\omega^*$ has this property. This was proved in 

>> E. van Douwen, K. Kunen, and J. van Mill, "There can be $C^*$-embedded dense proper subspaces of $\beta\omega - \omega$," *Proc. Amer. Math. Soc.* **105** (1989), pp. 462-470, available [here][1].

However, Fine and Gillman proved that $\omega^*$ consistently does not have this property (CH implies that it does not).

I do not know whether your question has a positive answer in ZFC (but I suspect that it does).

[Here][2] is a link to a related paper that you might find interesting (it's where I learned about the two results I mention above).


  [1]: https://staff.fnwi.uva.nl/j.vanmill/papers/papers1989/canbe.pdf
  [2]: http://arxiv.org/pdf/1310.0678v3.pdf