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We would recommend an electrician evaluate the lighting, and possible replacement of the fluorescent light as its filed might damage ESD sensitive items.
Regarding the mat, we are not aware of any phenomenon where the resistance of an ESD mat would change due to a fluorescent light inducing a charge. It is more likely to interfere with the measuring device, the Megohmmeter. To test this theory, you could shield the Megohmmeter by enclosing in an ESD shielding test, and seeing if an accurate resistance measurement can be taken.
Regarding powered equipment, per ESD Handbook TR20.20 section 5.5.4 Equipment Caused EOS (Electrical Overstress) "Electrical transients generated by large inductors, e.g. solenoids, motors, relays, faulty test equipment, and faulty grounds, can cause EOS. Grounding does not guarantee protection from such transients. Equipment that has large inductors should have protection devices to prevent the electrical 'spikes' (transients) being induced into ESD sensitive hardware. To measure the electrical spikes, use a storage oscilloscope connected to the part of the machine that comes in contact with the hardware."