A general model for a two-armed bandit with delayed responses is introduced and solved with dynamic programming. One arm has geometric lifetime with parameter $\theta$, which has prior distribution $\mu$. The other arm has known lifetime with mean $\kappa$. The response delays completely change the character of the optimal strategies from the no delay case; in particular, the bandit is no longer a stopping problem. The delays also introduce an extra parameter $p$ into the state space. In clinical trial applications, this parameter represents the number of patients previously treated with the unknown arm who are still living. The value function is introduced and investigated as $p, \mu$ and $\kappa$ vary. Under a regularity condition on the discount sequence, there exists a manifold in the state space such that both arms are optimal on the manifold, arm $x$ is optimal on one side and arm $y$ on the other. Properties of the manifold are described.