  The KATRIN experiment aims to measure the effective electron antineutrino
mass $m_{\overline{\nu}_e}$ with a sensitivity of 0.2 eV/c$^2$ using a gaseous
tritium source combined with the MAC-E filter technique. A low background rate
is crucial to achieving the proposed sensitivity, and dedicated measurements
have been performed to study possible sources of background electrons. In this
work, we test the hypothesis that gamma radiation from external radioactive
sources significantly increases the rate of background events created in the
main spectrometer (MS) and observed in the focal-plane detector. Using detailed
simulations of the gamma flux in the experimental hall, combined with a series
of experimental tests that artificially increased or decreased the local gamma
flux to the MS, we set an upper limit of 0.006 count/s (90% C.L.) from this
mechanism. Our results indicate the effectiveness of the electrostatic and
magnetic shielding used to block secondary electrons emitted from the inner
surface of the MS.
