  Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime of
good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field
satisfying a modified "Heegner hypothesis" in which $p$ splits, write
$K_\infty$ for the anticyclotomic $\mathbb Z_p$-extension of $K$ and let
$\Lambda$ denote the Iwasawa algebra of $K_\infty/K$. By extending to the
supersingular case the $\Lambda$-adic Kolyvagin method originally developed by
Bertolini in the ordinary setting, we prove that Kobayashi's plus/minus
$p$-primary Selmer groups of $E$ over $K_\infty$ have corank $1$ over
$\Lambda$. As an application, when all the primes dividing the conductor of $E$
split in $K$, we combine our main theorem with results of \c{C}iperiani and of
Iovita-Pollack and obtain a "big O" formula for the $\mathbb Z_p$-corank of the
$p$-primary Selmer groups of $E$ over the finite layers of $K_\infty/K$ that
represents the supersingular counterpart of a well-known result for ordinary
primes.
