  Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction
at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating
$p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a
large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional
zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More
generally, let $f$ be the weight-two newform associated with $A$, let
$f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the
Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\infty}$.
We prove a $p$-adic Gross$-$Zagier formula, expressing the quadratic term of
the Taylor expansion of $L_{p}(f_{\infty},k,s)$ at $(k,s)=(2,1)$ as a non-zero
rational multiple of the extended height-weight of a Heegner point in
$A(\mathbb{Q})$.
