Two-particle angular correlations are studied in proton-lead collisions at a nucleon-nucleon centre-of-mass energy of sNN‾‾‾‾√=5TeV, collected with the LHCb detector at the LHC. The analysis is based on data recorded in two beam configurations, in which either the direction of the proton or that of the lead ion is analysed. The correlations are measured as a function of relative pseudorapidity, Δη, and relative azimuthal angle, Δϕ, for events in different classes of event activity and for different bins of particle transverse momentum. In high-activity events a long-range correlation on the near side, Δϕ≈0, is observed in the pseudorapidity range 2.0<η<4.9. This measurement of long-range correlations on the near side in proton-lead collisions extends previous observations into the forward region up to η=4.9. The correlation increases with growing event activity and is found to be more pronounced in the direction of the lead beam. However, the correlation strengths in the direction of the lead and proton beams are found to be compatible when comparing events with similar absolute activity in the direction analysed.