  The local theory of complex dimensions describes the oscillations in the
geometry (spectra and dynamics) of fractal strings. Such geometric oscillations
can be seen most clearly in the explicit volume formula for the tubular
neighborhoods of a $p$-adic fractal string $\mathcal{L}_p$, expressed in terms
of the underlying complex dimensions. The general fractal tube formula obtained
in this paper is illustrated by several examples, including the nonarchimedean
Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a
$p$-adic fractal string coincides with the abscissa of convergence of the
geometric zeta function associated with the string, as well as with the
asymptotic growth rate of the corresponding geometric counting function. The
proof of this new result can be applied to both real and $p$-adic fractal
strings and hence, yields a unifying explanation of a key result in the theory
of complex dimensions for fractal strings, even in the archimedean (or real)
case.
