An overlapping-generations model with income heterogeneity is developed to analyze the impact of introducing a Central Bank Digital Currency (CBDC) on financial inclusion, and its potential adverse effect on bank funding. We highlight the role of two design parameters: the fixed cost of CBDC usage and the interest rate it pays, and derive principles for maximum inclusion and for mitigating the inclusion-intermediation trade-off. Agents' choice of money instrument is endogenously driven by income heterogeneity. Pre-CBDC, wealthier agents adopt deposits, while poorer agents adopt cash and remain unbanked. CBDCs with low fixed costs (and low interest rates) are adopted by cash holders and directly increase inclusion. CBDCs with high fixed costs (and high interest rates) are adopted by deposit holders and increase inclusion by raising deposit rates. The former allows for more favorable inclusion-intermediation trade-offs. We calibrate the model to match the US income distribution and aggregate share of unbanked households. A CBDC 50% cheaper (30% more expensive) than bank deposits decreases financial exclusion by 93% (71%) without impacting intermediation. In comparison, making the deposit market perfectly competitive would only decrease exclusion by 45%.