The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any closed convex bounded subset $C\subseteq X $ and any nonexpansive map $T:C\to C $ there is a $x\in C$ such that $T(x)=x$)

The space $X$ isn't uniformly convex so I can't use theorems about uniformly convex. There is no theorems about F.P.P in products of spaces and I have no another idea. Does someone have any idea?