Existence of a Hölder-free space

As AIM_BLB says in the comments, every Holder space of exponent $\alpha < 1$ is a Lipschitz space with respect to the metric $d^\alpha$. So the answer is an immediate "yes". May I add that I discuss Holder spaces at length in my book Lipschitz Algebras (2nd edition).

Kalton [Collect. Math. 55 (2004), no. 2, 171–217] studied several versions of such spaces, see the definitions on page 180. This paper was reprinted in Nigel J. Kalton selecta. Vol. 2. Birkhäuser/Springer, 2016.