We report on a search for eleven lepton-number violating processes B⁺ → X⁻ ℓ⁺ℓ′⁺ with X⁻ = K⁻, π⁻, ρ⁻, K*⁻, or D⁻ and ℓ⁺/ℓ′⁺ = e⁺ or μ⁺, using a sample of 471 ± 3 million BB̅ events collected with the BABAR detector at the PEP-II e⁺e⁻ collider at the SLAC National Accelerator Laboratory. We find no evidence for any of these modes and place 90% confidence level upper limits on their branching fractions in the range (1.5-26) x 10⁻⁷.