  Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions
of imaginary quadratic fields has been studied by several authors, starting
from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial
assumption that the prime $p$ is unramified in $K$. We start in this article
the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is
ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached
to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and
$p$ is ramified in $K$, we show an analogue of the exceptional zeroes
phenomenon investigated by Bertolini-Darmon in the setting when $p$ is inert in
$K$. More precisely, we consider situations in which the $p$-adic $L$-function
$L_p(E/K)$ of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ does
not vanish identically but, by sign reasons, has a zero at certain characters
$\chi$ of the Hilbert class field of $K$. In this case we show that the value
at $\chi$ of the first derivative of $L_p(E/K)$ is equal to the formal group
logarithm of the specialization at $p$ of a global point on the elliptic curve
(actually, this global point is a twisted sum of Heegner points). This
generalizes similar results of Bertolini-Darmon, available when $p$ is inert in
$K$ and $\chi$ is the trivial character.
