  We first show the realization of exceptional points in a non-Hermitian
superconducting system based on a conventional superconductor and then
demonstrate that, surprisingly, the system hosts odd-frequency pairing, solely
generated by the non-Hermiticity. While there is a coexistence of even- and
odd-frequency pairs under general conditions, we find that the even-frequency
term vanishes at the exceptional degeneracies, leaving only odd-frequency
pairing. This exceptional odd-frequency pairing is directly given by the
imaginary part of the eigenvalues at the exceptional points and can be measured
from the spectral function. Our results thus put forward non-Hermitian systems
as a powerful platform to realize odd-frequency superconducting pairing.
