Lipschitz free spaces isomorphic to their infinite sums and geometric applications

We find general conditions under which Lipschitz-free spaces over metric spaces are isomorphic to their infinite direct ${\mathrm{\ell }}_1$-sum and exhibit several applications. As examples of such applications we have that Lipschitz-free spaces over balls and spheres of the same finite dimensions are isomorphic, that the Lipschitz-free space over ${\mathbb{Z}}^d$ is isomorphic to its ${\mathrm{\ell }}_1$-sum, or that the Lipschitz-free space over any snowflake of a doubling metric space is isomorphic to ${\mathrm{\ell }}_1$. Moreover, following new ideas from [E. Bru`e, S. Di Marino and F. Stra, Linear Lipschitz and $C^1$ extension operators through random projection, arXiv:1801.07533] we provide an elementary self-contained proof that Lipschitz-free spaces over doubling metric spaces are complemented in Lipschitz-free spaces over their superspaces and they have BAP. Everything, including the results about doubling metric spaces, is explored in the more comprehensive setting of $p$-Banach spaces, which allows us to appreciate the similarities and differences of the theory between the cases $p<1$ and $p=1$.