On closed balls of a subset of a metric space

Let $X$ be a vector space and let $A$ be a subset of $X$. Let $d$ be a metric on $X$ which is not induced by a norm. Let $ain A$ and $r>0$. Let $B_A(a,r):={xin A: d(a,x)<r}$ and $B_A[a,r]:={xin A: d(a,x)leq r}$. Is it true that $B_A[a,r]subset overline{B(0,1)+{a}}^d$ (where $B(0,1)={xin X: d(x,0)<1}$) ? Or, $B_A[a,r]subset overline{B(0,1)+A}^d$ ?