Hi, I work with an eye surgeon once a week giving anaesthetics. Most likely removing a lesion from the conjunctiva will be done under local, unless you have an anxiety disorder or a movement disorder. (Like Parkinson's) Typically you have numbing drops placed first which sting a little bit, but then numb the surface layer of the eyeball. Then the surgeon will supplement the drops with an injection of local to minimise bleeding. The weirdest bit is the eyelid retractor that holds your eye open for the procedure. It all sounds heinous but it is not that bad, we do cataracts under local and most of my patients range in age between 50-100 years, and even the querulous oldies cope really well. Removing a lesion takes about 20 mins or so.
Good luck, best to get it taken care of, you can get melanomas in your eye.
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Me 37 DH 50
DD 2006
2x Angels 2010
IVF/ICSI May 2011 BFP