In the last years, several empirical approaches have been proposed to tackle argument mining tasks, e.g., argument classification, relation prediction, argument synthesis. These approaches rely more and more on language models (e.g., BERT) to boost their performance. However, these language models require a lot of training data, and size is often a drawback of the available argument mining data sets. The goal of this paper is to assess the robustness of these language models for the argument classification task. More precisely, the aim of the current work is twofold: first, we generate adversarial examples addressing linguistic perturbations in the original sentences, and second, we improve the robustness of argument classification models using adversarial training. Two empirical evaluations are addressed relying on standard datasets for AM tasks, whilst the generated adversarial examples are qualitatively evaluated through a user study. Results prove the robustness of BERT for the argument classification task, yet highlighting that it is not invulnerable to simple linguistic perturbations in the input data.