  The aim of this paper is to provide an analogue of the Ball-Rivoal theorem
for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a
Dirichlet character $\chi$ and a number field $K$ the formula
$\dim_{K}(K+\sum_{i=2}^{s+1} L_p(i,\chi\omega^{1-i}) K )\geq
\frac{(1-\epsilon)\log (s)}{2[K:\mathbb{Q}](1+\log 2)}$. As a byproduct, we
establish an asymptotic linear independence result for the values of the
$p$-adic Hurwitz zeta function.
