Anticyclotomic $p$-adic $L$-functions for elliptic curves at some additive reduction primes

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 $\text{L}$ attached to $E\mathrm{/}K$ when $E\mathrm{/}{\text{QQ}}_p$ attains semistable reduction over an abelian extension. We prove that $\text{L}$ satisfies the expected interpolation properties; namely, we show that if $\chi$ is an anticyclotomic character of conductor $c{p}^n$ then $\chi (\text{L})$ is equal (up to explicit constants) to $L(E,\chi ,1)$ or $L^{\mathrm{\prime }}(E,\chi ,1)$.