  We find general conditions under which Lipschitz-free spaces over metric
spaces are isomorphic to their infinite direct $\ell_1$-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
$\ell_1$-sum, or that the Lipschitz-free space over any snowflake of a doubling
metric space is isomorphic to $\ell_1$. 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 $p=1$.
