ALICE is one of four large experiments at the CERN Large Hadron Collider, specially designed to study particle production in ultra-relativistic heavy-ion collisions. Located 52 meters underground with 28 meters of overburden rock, it has also been used to detect muons produced by cosmic-ray interactions in the upper atmosphere. The large size and excellent tracking capability of the ALICE Time Projection Chamber are exploited to study the muonic component of extensive air showers. We present the multiplicity distribution of these atmospheric muons and its comparison with Monte Carlo simulations. The latest version of the QGSJET hadronic interaction model was used to simulate the development of the resulting air showers. High multiplicity events containing more than 100 reconstructed muons were also studied. Similar events have been studied in previous underground experiments such as ALEPH and DELPHI at LEP without satisfactory explanations for the frequency of the highest multiplicity events. We demonstrate that the high muon-multiplicity events observed in ALICE stem from primary cosmic rays with energies above 1016 eV and that the frequency of these events can be successfully described by assuming a heavy mass composition of primary cosmic rays in this energy range.