A measurement of the production of prompt Λ _c ⁺ baryons in Pb–Pb collisions at s _NN =5.02 TeV with the ALICE detector at the LHC is reported. The Λ _c ⁺ and Λ‾ _c ⁻ were reconstructed at midrapidity (|y|<0.5) via the hadronic decay channel Λ _c ⁺ →pK _S ⁰ (and charge conjugate) in the transverse momentum and centrality intervals 6<p _T <12 GeV/c and 0–80%. The Λ _c ⁺ /D ⁰ ratio, which is sensitive to the charm quark hadronisation mechanisms in the medium, is measured and found to be larger than the ratio measured in minimum-bias pp collisions at s=7 TeV and in p–Pb collisions at s _NN =5.02 TeV. In particular, the values in p–Pb and Pb–Pb collisions differ by about two standard deviations of the combined statistical and systematic uncertainties in the common p _T interval covered by the measurements in the two collision systems. The Λ _c ⁺ /D ⁰ ratio is also compared with model calculations including different implementations of charm quark hadronisation. The measured ratio is reproduced by models implementing a pure coalescence scenario, while adding a fragmentation contribution leads to an underestimation. The Λ _c ⁺ nuclear modification factor, R _AA , is also presented. The measured values of the R _AA of Λ _c ⁺ , D _s ⁺ and non-strange D mesons are compatible within the combined statistical and systematic uncertainties. They show, however, a hint of a hierarchy (R _AA ^{D 0} <R _AA ^{D _s +} <R _AA ^{Λ _c +} ), conceivable with a contribution from coalescence mechanisms to charm hadron formation in the medium.