We have a fault in the cabling for one of our kitchen ceiling lights.
The lamp I have just removed was running rather warm, and it's cooked
the end of the cable.

I can't replace the cable without taking up carpets and loads of
floorboards upstairs. And I've pulling on the cable, but it won't
budge, so there's no hope of pulling a new one through.

My plan is to trace the cable back 30 cms to a sound bit of cable, make
a hole in the ceiling, and extend the cable from there to the lamp. The
idea is to put the replacement light in the same position as the old one.

So, some questions:

Are there any connectors I can use that are allowed within the regulations?

Are there any connectors I can use that are safe, even though not
allowed within the regulations?

I'd prefer a small connector, so I don't have to make a whopping hole in
the ceiling. Any suggestions, please?