Not at the levels we like to use for surgery, certainly. But my understanding is that a “medically-induced coma” (the potentially-irreversible palliative-care kind)
does suppress all pain signalling. Likewise, people in a vegetative state have no physiological response to pain. If you
completely suppress central functioning, then you
completely suppress pain response.
(There might still be a local release of pro-inflammatory cell-danger-response purines from the wound site that do things when they hit various organs/tissues on their way through the circulation, but I believe we don’t tend to call those “pain signals”, for the same reason we don’t call them that in plants or fungi.)