We introduce a new class of lattice models based on a continuous time Markov chain approximation scheme for affine processes, whereby the approximating process itself is affine. A key property of this class of lattice models is that the location of the time nodes can be chosen in a payoff dependent way and one has the flexibility of setting them only at the relevant dates. Time stepping invariance relies on the ability of computing node-to-node discounted transition probabilities in analytically closed form. The method is quite general and far reaching and it is introduced in this article in the framework of the broadly used single-factor, affine short rate models such as the Vasiček and CIR models. To illustrate the use of affine lattice models in these cases, we analyze in detail the example of Bermuda swaptions.