  In this paper we present part I of nonlinear operator ideals theory between
metric spaces and Banach spaces. Building upon the definition of operator ideal
between arbitrary Banach spaces of A. Pietsch we pose three types of nonlinear
versions of operator ideals. We introduce several examples of nonlinear ideals
and the relationships between them. For every space ideal $\mathsf{A}$ can be
generated by a special nonlinear ideal which consists of those Lipschitz
operators admitting a factorization through a Banach space
$\mathbf{M}\in\mathsf{A}$. We investigate products and quotients of nonlinear
ideals. We devote to constructions three types of new nonlinear ideals from
given ones. A "new" is a rule defining nonlinear ideals
$\mathfrak{A}^{L}_{new}$, $\textswab{A}^{L}_{new}$, and
$\textfrak{A}^{L}_{new}$ for every $\mathfrak{A}$, $\textswab{A}^{L}$, and
$\textfrak{A}^{L}$ respectively, are called a Lipschitz procedure. Considering
the class of all stable objects for a given Lipschitz procedure we obtain
nonlinear ideals having special properties. We present the concept of a
(strongly) $p-$Banach nonlinear ideal ($0<p<1$) and prove that the nonlinear
ideals of Lipschitz nuclear operators, Lipschitz Hilbert operators, products
and quotient are strongly $r-$Banach nonlinear ideals ($0<r<1$).
