Using muon-spin rotation the pressure-induced superconductivity in the Bi-III phase of elemental bismuth (transition temperature T_c ≃ 7.05 K) was investigated. A Ginzburg-Landau parameter κ = λ/ξ = 30(6) (λ is the magnetic penetration depth, ξ is the coherence length) was estimated, which turns out to be the highest among known single element superconductors. The temperature dependence of the superconducting energy gap [Δ(T)] reconstructed from λ^-2(T) deviates from the weakly coupled BCS prediction. The coupling strength 2Δ/k_BT_c ≃ 4.34 was estimated, thus implying that Bi-III stays within the strong-coupling regime. The density functional theory calculations suggest that superconductivity in Bi-III could be described within the Eliashberg approach with a characteristic phonon frequency ω_ln ≃ 5.5 meV. An alternative pairing mechanism to the electron-phonon coupling involves the possibility of Cooper pairing induced by Fermi-surface nesting.