  Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive
reduction. Let $K$ be an imaginary quadratic field and fix a positive integer
$c$ prime to the conductor of $E$. The main goal of the present article is to
define an anticyclotomic $p$-adic $L$-function $\L$ attached to $E/K$ when
$E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that
$\L$ satisfies the expected interpolation properties; namely, we show that if
$\chi$ is an anticyclotomic character of conductor $cp^n$ then $\chi(\L)$ is
equal (up to explicit constants) to $L(E,\chi,1)$ or $L'(E,\chi,1)$.
