Discrete breathers can exist in nonlinear lattices as exact time-periodic solutions only in the absence of perturbations. In reality, long-lived spatially localized vibrational modes, called quasi-breathers (QBs), are of interest because they contribute to the macroscopic properties of crystals. Crystals with complex structure may have a gap in the phonon spectrum, and gap QBs may exist, vibrating at frequencies in the gap. Here, a binary crystal with B2 (CsCl) structure is analyzed taking into account interactions up to the fourth neighbor. The atoms of two spices have masses 𝑚1 and 𝑚2. The pairwise interatomic interactions with hard or soft cubic anharmonicity are considered. The conditions on the mass ratio 𝑚2∕𝑚1 and the stiffness of the interatomic bonds are derived under which the gap appears in the phonon spectrum and short- or long-lived gap QBs can exist. Short-lived QBs have the main frequency in the gap, but higher harmonics fall in the optical band of the phonon spectrum. Long-lived QBs have the main frequency and all higher harmonics outside the phonon spectrum. Examples of numerically found gap QBs are given. Our results provide a theoretical background for finding gap QBs in B2 crystals with components whose difference in atomic mass is large enough to open a gap in the phonon spectrum; examples of such crystals are given.