A search for 5.5-MeV solar axions produced in the $p+d→\rm{^3He}+A (5.5 \rm{MeV})$ reaction was performed using the Borexino detector. The Compton conversion of axions to photons, ${\rm A}+e→ e+γ$; the axio-electric effect, ${\rm A}+e+Z→ e+Z$; the decay of axions into two photons, ${\rm A}→2γ$; and inverse Primakoff conversion on nuclei, ${\rm A}+Z→γ+Z$, are considered. Model independent limits on axion-electron ($g_{Ae}$), axion-photon ($g_{Aγ}$), and isovector axion-nucleon ($g_{3AN}$) couplings are obtained: $|g_{Ae}\times g_{3AN}| ≤ 5.5\times 10^{-13}$ and $|g_{Aγ}\times g_{3AN}| ≤ 4.6\times 10^{-11} \rm{GeV}^{-1}$ at $m_A