This paper investigates the problem of secret key generation from correlated Gaussian random variables in the short blocklength regime. Short blocklengths are commonly employed in massively connected IoT sensor networks in 5G and beyond wireless systems. Polar codes have previously been shown to be applicable to the secret key generation problem, and are known to perform well for short blocklengths in the channel coding context. Inspired by these findings, we propose an explicit protocol based on polar codes for generating secret keys in the short blocklength regime. This protocol differs from previously proposed key generation protocols based on polar coding in two main ways: (i) we consider a Gaussian source for the key generation; (ii) we focus on the short blocklength regime. Simulation results show that the proposed protocol performs well even for very short blocklengths, especially if one can relax the BER/BLER requirements for the generated keys. They also demonstrate that the polar code based protocol outperforms a similar one using LDPC codes in place of polar codes, and that this advantage grows the shorter the blocklength becomes.