  The notion of $p$-summing Bloch mapping from the complex unit open disc
$\mathbb{D}$ into a complex Banach space $X$ is introduced for any $1\leq
p\leq\infty$. It is shown that the linear space of such mappings, equipped with
a natural seminorm $\pi^{\mathbb{B}}_p$, is M\"obius-invariant. Moreover, its
subspace consisting of all those mappings which preserve the zero is an
injective Banach ideal of normalized Bloch mappings. Bloch versions of the
Pietsch's domination/factorization Theorem and the Maurey's extrapolation
Theorem are presented. We also introduce the spaces of $X$-valued Bloch
molecules on $\mathbb{D}$ and identify the spaces of normalized $p$-summing
Bloch mappings from $\mathbb{D}$ into $X^*$ under the norm $\pi^{\mathbb{B}}_p$
with the duals of such spaces of molecules under the Bloch version of the
$p$-Chevet--Saphar tensor norms $d_p$.
