Quantifications of in-situ denudation rates on vertical headwalls, averaged over millennia, have been thwarted because of inaccessibility. Here, we benefit from a tunnel crossing a large and vertical headwall in the European Alps (Eiger), where we measured concentrations of in-situ cosmogenic Cl-36 along five depth profiles linking the tunnel with the headwall surface. Isotopic concentrations of Cl-36 are low in surface samples, but high at depth relative to expectance for their position. The results of Monte-Carlo modelling attribute this pattern to inherited nuclides, young minimum exposure ages and to fast average denudation rates during the last exposure. These rates are consistently high across the Eiger and range from 45 +/- 9 cm kyr(-1) to 356 +/- 137 cm kyr(-1) (1 sigma) for the last centuries to millennia. These high rates together with the large inheritance point to a mechanism where denudation has been accomplished by frequent, cm-scale rock fall paired with chemical dissolution of limestone.