Algorithmic tools are often used to make decisions about people in high-stakes domains. In the presence of such automated decision making, there is incentive for strategic agents to modify their input to the algorithm in order to receive a more desirable outcome. While previous work on strategic classification attempts to capture this phenomenon, these models fail to take into account the multiple actions a decision maker usually has at their disposal, and the fact that they often have access only to bandit feedback. In contrast, we capture this setting as a contextual bandit problem, in which a decision maker must take actions based on a sequence of strategically modified contexts. We provide a low-strategic-regret algorithm for the two action setting, and prove that sublinear strategic regret is generally not possible for settings in which the number of actions is greater than two. Along the way, we obtain impossibility results for multi-class strategic classification which may be of independent interest.