Stealing [an answer from "user940" on Math.SE](https://math.stackexchange.com/questions/2022647/equality-of-two-borel-measures), the answer is yes, such a measures can exist, at least for closed balls.  In the paper

> <cite authors="Davies, Roy O.">_Davies, Roy O._, [**Measures not approximable or not specifiable by means of balls**](http://dx.doi.org/10.1112/S0025579300005386), Mathematika, Lond. 18, 157-160 (1971). [ZBL0229.28005](https://zbmath.org/?q=an:0229.28005).</cite>

the author constructs a compact metric space $X$ and two distinct Borel probability measures $\mu_1, \mu_2$ that agree on every closed ball.  (They must therefore also agree on open balls, because an open ball is a countable increasing union of closed balls.)  Taking $\mu = \mu_1 - \mu_2$ provides your desired signed measure.