Shave them with a razor.  That removes them quite effectively.  Not an
electric, the blade kind.

Den said:

Love the flowers, hate having to weed or prune them.  When I get a lot of
stickers in me I use either packaging tape or duct tape.  Lay a piece on, pull it
off and toss it.  Lay on a fresh piece and repeat.

A sharp tweezer with a built in magnifying lens is useful in hunting down stray
prickles.

But I always get a few spines that get in beyond either remedy and I have to
wait for these to fester out.

I don't have a whole lot of experience pruning prickly pear cacti here
in Ontario Canada but I was sent some as gifts by friends in the SW US
along with instructions to use newspaper to handle the things, if
handling was necessary.

They've overwintered here for almost three years now, producing copious
amounts of amazingly beautiful flowers and fruit.

In any case, on the issue of mico-spines embedded in flesh:

Years ago when I didn't know any better and used fibreglass insulation,
it would invariably leave irritating fibres embedded in all exposed
flesh. Slathering on a layer of woodworkers glue, waiting for it to
skin and then peeling off the skin would usually do the trick.

The same might work to remove the cacti pricks.