In this paper, the problem of network connectivity is studied for an adversarial Internet of Battlefield Things (IoBT) system in which an attacker aims at disrupting the connectivity of the network by choosing to compromise one of the IoBT nodes at each time epoch. To counter such attacks, an IoBT defender attempts to reestablish the IoBT connectivity by either deploying new IoBT nodes or by changing the roles of existing nodes. This problem is formulated as a dynamic multistage Stackelberg connectivity game that extends classical connectivity games and that explicitly takes into account the characteristics and requirements of the IoBT network. In particular, the defender's payoff captures the IoBT latency as well as the sum of weights of disconnected nodes at each stage of the game. Due to the dependence of the attacker's and defender's actions at each stage of the game on the network state, the feedback Stackelberg solution (FSE) is used to solve the IoBT connectivity game. Then, sufficient conditions under which the IoBT system will remain connected, when the FSE solution is used, are determined analytically. Numerical results show that the expected number of disconnected sensors, when the FSE solution is used, decreases up to 62% compared to a baseline scenario in which a Stackelberg game with no feedback is used, and up to 57% compared to a baseline equal probability policy.