Here is a counterexample constructed from the $5$-point example given by Włodzimierz Holsztyński here.

Consider the set $Z = \{x,y,a,b,c\}$ made into a metric space via
$$d(x\ y) = d(a\ b) = 1,$$
$$d(x\ a) = d(y\ b) = 2,$$

$$d(x\ b) = d(y\ a) = 3,$$

$$d(x\ c) = d(y\ c) = 6,$$
$$d(a\ c) = 5,\qquad\qquad d(b\ c) = 4.$$

First we construct a connected metric space $Z'$ simply by connecting each two points by edges of lengths given by the respective distances and taking the metric on $Z'$ the induced intrinsic one (i hope this is clear). Now we form a noncompact space $Z''$ by adding a ray (that is the interval $[0,\infty[$) starting at $c$. Finally consider the subset $X \subset Z''$ given by $Z''$ without the edge connecting $a$ to $c$ and the one connecting $b$ to $c$. Equip $X$ with the induced restricted metric $d$ (not the intrinsic one). Then for all $r > 0$ the balls $B_r(x) \subset X$ and $B_r(y) \subset X$ are isometric. Therefore, taking $X_i = X$ and $p_i = x$ we find that $B_r(p_i) \to B_r(p)$ in Gromov-Hausdorff sense for all $r > 0$, but $(B_r(p_i),p_i) = (B_r(x),x)$ does not converge to $(B_r(p),p) = (B_r(y),y)$ for all $r \geq 6$ since there is no isometry of $X$ taking $x$ to $y$.

Maybe a sufficient condition might be that the metrics of the $X_i$ are intrinsic (and complete), i.e. distances are given by infimal (minimal) lengths of continuous curves connecting points.