Find a sequence of disjoint balls $(B_n)$, all contained in the unit ball of $X$ and all having radius greater than some fixed $\epsilon$. Select continuous functions $f_n: B_n \to X$ however you like, such that each $f_n$ goes to zero on the boundary of $B_n$. Then patch together to get a continuous function on $\bigcup B_n$, and extend by zero elsewhere to get a function from $X$ to $X$. There needn't be a uniform bound on the $f_n$ for this to work.

Find a sequence of disjoint balls $(B_n)$, all contained in the unit ball of $X$ and all having radius greater than some fixed $\epsilon$. Select continuous functions $f_n: B_n \to X$ however you like, such that each $f_n$ is zero outside radius $\epsilon/2$. Then patch together to get a continuous function on $\bigcup B_n$, and extend by zero elsewhere to get a function from $X$ to $X$. There needn't be a uniform bound on the $f_n$ for this to work.