$x_n \in B(p,e^{-n\delta})$ iff $p \in B(x_n,e^{-n\delta})$. Thus we ask whether $$ \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty B(x_n,e^{-n\delta}) = \varnothing $$

$x_n \in B(p,e^{-n\delta})$ iff $p \in B(x_n,e^{-n\delta})$. Thus we ask whether $$ \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty B(x_n,e^{-n\delta}) = \varnothing $$ But that is a countable intersection of dense open sets, so (by Baire category) is NOT empty.

(I hope my quantifiers are right...)