A search for $W^\prime$ bosons in events with one lepton (electron or muon) and missing transverse momentum is presented. The search uses 3.2 fb$^{-1}$ of $pp$ collision data collected at $\sqrt{s} = 13$ TeV by the ATLAS experiment at the LHC in 2015. The transverse mass distribution is examined and no significant excess of events above the level expected from Standard Model processes is observed. Upper limits on the $W^\prime$ boson cross-section times branching ratio to leptons are set as a function of the $W^\prime$ mass. Assuming a $W^\prime$ boson as predicted by the Sequential Standard Model, $W^\prime$ masses below 4.07 TeV are excluded at the 95% confidence level. This extends the limit set using LHC data at $\sqrt{s}=8$ TeV by around 800 GeV.