We establish fundamental upper bounds on the amount of secret key that can be extracted from quantum Gaussian states by using only local Gaussian operations, local classical processing, and public communication. For one-way public communication, or when two-way public communication is allowed but Alice and Bob first perform destructive local Gaussian measurements, we prove that the key is bounded by the R\'enyi-$2$ Gaussian entanglement of formation $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$. Since the inequality is saturated for pure Gaussian states, this yields an operational interpretation of the R\'enyi-$2$ entropy of entanglement as the secret key rate of pure Gaussian states that is accessible with Gaussian operations and one-way communication. In the general setting of two-way communication and arbitrary interactive protocols, we argue that $2 E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ is still an upper bound on the extractable key. We conjecture that the factor of $2$ is spurious, which would imply that $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ coincides with the secret key rate of Gaussian states under Gaussian measurements and two-way public communication. We use these results to prove a gap between the secret key rates obtainable with arbitrary versus Gaussian operations. Such a gap is observed for states produced by sending one half of a two-mode squeezed vacuum through a pure loss channel, in the regime of sufficiently low squeezing or sufficiently high transmissivity. Finally, for a wide class of Gaussian states that includes all two-mode states, we prove a recently proposed conjecture on the equality between $E_{F,2}^{\mathrm{\scriptscriptstyle G}}$ and the Gaussian intrinsic entanglement. The unified entanglement quantifier emerging from such an equality is then endowed with a direct operational interpretation as the value of a quantum teleportation game.

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