Complemented Banach spaces. Let $X$ be  Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties:

$Z\cap Y=\{0\}$ and every $x\in X$  can be written in a unique form as $x=y+z$ with $y\in Y$ and $z\in Z$

Can we conclude that $Y$ is complemented in $X$?

Edit: I'm not asking if $Z$ is a complement of $Y$ in $X$. Indeed, $Z$ does not need to be linear. What I am asking is if we can find a set closed linear $W\subset X$ such that $W$ is a complement of $Y$ in $X$.
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Take $X = \mathbb R^2$ with your favorite norm (say $l^2$ norm).  Let $Y$ be the $y$-axis, a closed linear subspace.  Let $Z$ be the graph of a continuous function $h$ with $h(0)=0$.  Say $h(x) = x^3$.  Every point of $X$ is uniquely the sum of a point on the graph plus a point of $Y$.  SO ... the set $Z$ need not be the complement.  

Can we imitate this using $X$ a Banach space and $Y$ a non-complemented subspace?  Make a function $h$ choosing one element from each equivalence class?  

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phrase this variant in a different way ...
If $T : X \to U$ is a surjective continuous linear map of Banach spaces, and if there is a continuous section (that is, a continuous $V : U \to X$ with $V(T(u))=u$ for all $u$), then must there exist also a continuous linear section?
comment

Take $X = \mathbb R^2$ with your favorite norm (say $l^2$ norm).  Let $Y$ be the $y$-axis, a closed linear subspace.  Let $Z$ be the graph of a continuous function $h$ with $h(0)=0$.  Say $h(x) = x^3$.  Every point of $X$ is uniquely the sum of a point on the graph plus a point of $Y$.  SO ... the set $Z$ need not be the complement.  

Can we imitate this using $X$ a Banach space and $Y$ a non-complemented subspace?  Make a function $h$ choosing one element from each equivalence class?  

added
phrase this variant in a different way ...
If $T : X \to U$ is a surjective continuous linear map of Banach spaces, and if there is a continuous section (that is, a continuous $V : U \to X$ with $V(T(u))=u$ for all $u$), then must there exist also a continuous linear section?