We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif and Shirai and Takahashi. As its continuum version, we prove an isomorphism between the translation-invariant determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels and homogeneous Poisson point processes. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.