  We analyze the Hawking evaporation process of Reissner-Nordstr\"om black
holes. It is shown that the characteristic radiation quanta emitted by the
charged black holes may turn near-extremal black-hole spacetimes into
horizonless naked singularities. The present analysis therefore reveals the
intriguing possibility that the semi-classical Hawking evaporation process of
black holes may violate the fundamental Penrose cosmic censorship conjecture.
