  Let $A/\mathbb{Q}$ be an elliptic curve having split multiplicative reduction
at an odd prime $p$. Under some mild technical assumptions, we prove the
statement: $$rank_{\mathbb{Z}}A(\mathbb{Q})=1 \ \ and\ \ \
\#\Big(\underline{III}(A/\mathbb{Q})[p^\infty]\Big)<\infty\ \ \implies\ \
\mathrm{ord}_{s=1}L(A/\mathbb{Q},s)=1,$$ thus providing a "$p$-converse" to a
celebrated theorem of Kolyvagin-Gross-Zagier.
