How to flatten lumps in asphalt floor

An iron?

Seems likely to me that something evil lurks underneath, probably a bit of damp penetration.

Personally (and pls bear in mind I am not any kind of expert) I'd chisel out the asphalt as far as necessary to remove all the raised area. I'd repair any crack in the concrete subfloor, then overpaint the whole area with 2 coats of bitumen-based waterproofing gunk, and finally lay a new section of screed or levelling compound to bring it back up to the level.

talc over the top?

If not, drop dry cement down to fill it. Again you will need some mastic to seal it.

If you never noticed it before then you might not need to bother just hide it with some thick underlay. Trying to forget it's there will be harder.

The pictures might have been more use had a straight edge been laid across the top to show levels. Don't go chopping your floor up. Think!

This sounds closest to what I was thinking I should do, although my wife has asked if we could re-melt the asphalt somehow. Is that feasible, or completely stupid? lol

I doubt it's damp as there's no sign of damp anywhere else. The house was built in 1955... I wonder if the sub-floor would have been built on a waterproof membrane that might already be damaged, or damaged by me chiselling?

Thanks all for the replies.

I'd want to cut them out to see what was causing the heave, and take it from there.

We were somewhere around Barstow, on the edge of the desert, when the drugs began to take hold. I remember Stone Cold saying something like:

Umm... your house wasn't occupied by Fred West at any time, was it?