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There has been extensive evidence demonstrating that deep neural networks are vulnerable to adversarial examples, which motivates the development of defenses against adversarial attacks. Existing adversarial defenses typically improve model robustness against individual specific perturbation types (\eg, $\ell_{\infty}$-norm bounded adversarial examples). However, adversaries are likely to generate multiple types of perturbations in practice (\eg, $\ell_1$, $\ell_2$, and $\ell_{\infty}$ perturbations). Some recent methods improve model robustness against adversarial attacks in multiple $\ell_p$ balls, but their performance against each perturbation type is still far from satisfactory. In this paper, we observe that different $\ell_p$ bounded adversarial perturbations induce different statistical properties that can be separated and characterized by the statistics of Batch Normalization (BN). We thus propose Gated Batch Normalization (GBN) to adversarially train a perturbation-invariant predictor for defending multiple $\ell_p$ bounded adversarial perturbations. GBN consists of a multi-branch BN layer and a gated sub-network. Each BN branch in GBN is in charge of one perturbation type to ensure that the normalized output is aligned towards learning perturbation-invariant representation. Meanwhile, the gated sub-network is designed to separate inputs added with different perturbation types. We perform an extensive evaluation of our approach on commonly-used dataset including MNIST, CIFAR-10, and Tiny-ImageNet, and demonstrate that GBN outperforms previous defense proposals against multiple perturbation types (\ie, $\ell_1$, $\ell_2$, and $\ell_{\infty}$ perturbations) by large margins.