But as the heading is changing, turning inward each increment of time, dT, there is a radial velocity change (remember velocity is a vector which means as the direction of R changes, Vr changes); so that dVr/dT <> 0 <> Ar. There is the radial acceleration and Fr = mAr is the average centripetal force causing that acceleration.
The derivation of Ar = Vt^2/R (your v^2/r) is a bit messy. But it's based on the fact that it takes work to turn an object when because of inertia that object really wants to go straight. And that work is dQ = Fr dR = mAr dT where the centripetal force Fr is turning the object inward and imparts the radial acceleration Ar.
So that's the bottom line. Although your speed is constant in this problem, the direction is not. And as velocity is a vector with both speed and direction, there is acceleration. And in this case, it's all radial acceleration imparted by the centripetal force turning the object inward.