  We investigate the structure of the free $p$-convex Banach lattice
$FBL^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice
exists, and giving a convenient functional representation of it, we focus our
study on how properties of an operator $T:E\rightarrow F$ between Banach spaces
transfer to the associated lattice homomorphism
$\overline{T}:FBL^{(p)}[E]\rightarrow FBL^{(p)}[F]$. Particular consideration
is devoted to the case when the operator $T$ is an isomorphic embedding, which
leads us to examine extension properties of operators into $\ell_p$, and
several classical Banach space properties such as being a G.T. space. A
detailed investigation of basic sequences and sublattices of free Banach
lattices is provided. In addition, we begin to build a dictionary between
Banach space properties of $E$ and Banach lattice properties of $FBL^{(p)}[E]$.
In particular, we characterize the existence of lattice copies of $\ell_1$ in
$FBL^{(p)}[E]$ and show that $FBL[E]$ has an upper $p$-estimate if and only if
$id_{E^*}$ is $(q,1)$-summing ($\frac{1}{p}+\frac{1}{q}=1$). We also highlight
the significant differences between $FBL^{(p)}$-spaces depending on whether $p$
is finite or infinite. For example, we show that $FBL^{(\infty)}[E]$ is lattice
isometric to $FBL^{(\infty)}[F]$ whenever $E$ and $F$ have monotone finite
dimensional decompositions, while, on the other hand, when $p<\infty$ and $E^*$
is smooth, $FBL^{(p)}[E]$ determines $E$ isometrically.
