Equivalent definitions of Injective Banach Spaces

Equivalent definitions of Injective Banach Spaces

A Banach space $X$ is said to be reflexive if for all Banach spaces $W,Z$
with $W\subset Z$, and operators $T\in B(W,X)$, $T$ can be extended to all
of $Z$ with the same norm.
Equivalently, $X$ is injective if it is complemented by a norm $1$
projection in any Banach space containing it.



Labelling the first definition as $(1)$ and the second definition as
$(2)$, the proof for $(1)\Rightarrow (2)$ is brief.
If $X\subset Y$ for some Banach space $Y$, then applying $(1)$ to the
identity map on $X$ yields the projection.



The $(2)\Rightarrow (1)$ direction I have been stuck on for a couple of
days. Can anyone offer a hint? Thanks very much in advance!