We formulate a multivariable $p$-adic Birch and Swinnerton-Dyer conjecture for $p$-ordinary elliptic curves $A$ over number fields $K$. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case $K=\mathbf{Q}$ and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, under mild conditions, in the following situation: $K$ is imaginary quadratic, $A=E_K$ is the base change to $K$ of an elliptic curve over the rationals, and the rank of $A$ is either 0 or $1$.

The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over $\mathbf{Q}$, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank $1$, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) $p$-adic Gross–Zagier and Waldspurger formulas in families.