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In this paper, a fractional-order model for HIV with drug-resistance and CTL immune response is established. Two cases (subsystem with drug-sensitive and subsystem with drug-resistant) are considered. For both subsystems: firstly, the existence and uniqueness of the positive solution is proved; secondly, the sufficient conditions for the stability of the disease-free equilibrium are obtained; finally, some numerical simulations are performed to verify the theoretical results. After that, the main system is analyzed in a similar way. Calculation indicates that the basic reproduction number of the main system is the maximum value of the basic reproduction numbers of the corresponding subsystems. In addition, through numerical simulation, we know that the drug efficacy plays an important role in the treatment of HIV. • A fractional-order model for HIV with drug-resistance and CTL immune response is constructed. • The basic reproduction number of the main model is the maximum value of the corresponding two subsystems. • Through numerical simulations, the relationship of basic reproduction number with respect to fractional order was found. [ABSTRACT FROM AUTHOR]