The answer is YES. Indeed, consider a metrics $\ \rho_A\ $ in A, topologically equivalent to the induced topology from $\ (X\ d),\ $ and such that $\ (A\ \rho_A)\ $ is isometric to the standard unit interval $\ [0;1]\ $ with the euclidean distance. Thus there is a function $\ f:[0;1]\rightarrow A,\ $ such that
$$ \forall_{t\in[0;1]}\quad\rho_A(v\ \,f(t))\ \ =\ \ t$$
Hausdorff theorem provides a metrics $\ \rho\ $ in $\ X,\ $ topologically equivalent to $\ d,\ $ and such that $ \rho|A\times A\ =\ \rho_A.\ $ Thus a required retraction $\ r: X\rightarrow A\ $ can be given as follows:
$$\forall_{p\in X}\quad r(p)\ :=\ f(\rho(v\ p))$$
Done, that's it.