The answer is no for the Banach space $c_0$. Suppose $B(x_i,r_i)$ is a sequence of balls with $r_i\to 0$ and WLOG $x_i$ is supported in $[1,N_i]$ with $N_1<N_2<...$. Consider a point $x$ in $c_0$ whose $N_i+1$ coordinate is $2 r_i$.
I think the answer is no for any separable Banach space: IIRC, for any separable Banach space $X$ and any increasing sequence $E_n$ of finite dimensional subspaces and any sequence of positive $r_n\to 0$, there is a vector $x$ in $X$ s.t. the distance from $x$ to $E_n$ is larger than $r_n$ (in fact, even equal to $2r_n$ if $r_n$ is decreasing).