The fragmentation properties of jets containing b-hadrons are studied using charged B mesons in 139 fb⁻¹ of pp collisions at s = 13 TeV, recorded with the ATLAS detector at the LHC during the period from 2015 to 2018. The B mesons are reconstructed using the decay of B^± into J/ψK^±, with the J/ψ decaying into a pair of muons. Jets are reconstructed using the anti-k_t algorithm with radius parameter R = 0.4. The measurement determines the longitudinal and transverse momentum profiles of the reconstructed B hadrons with respect to the axes of the jets to which they are geometrically associated. These distributions are measured in intervals of the jet transverse momentum, ranging from 50 GeV to above 100 GeV. The results are corrected for detector effects and compared with several Monte Carlo predictions using different parton shower and hadronisation models. The results for the longitudinal and transverse profiles provide useful inputs to improve the description of heavy-flavour fragmentation in jets. [Figure not available: see fulltext.].