You can come close, using the segment muxer.
ffmpeg -i source_hls.m3u8 -c copy -f segment -segment_time 60 -segment_wrap 2 -reset_timestamps 1 out%02d.mkv -y
This will write to out00.mkv, then out01.mkv, then overwrite out00.mkv, next overwrite out01.mkv and so on.
The segment time is set at 60 seconds, so each segment will be around 60 seconds. The targets for splitting are 60,120,180,240... seconds of the input. However, video streams will be only be split at keyframes at or after the split target. So, if the first keyframe after t=59 is at 66, then the first segment will be 66s long. The next target is 120s. Let's say there's a KF at 121s, so the 2nd segment will be 66 to 121s = 55s long. Something to keep in mind when checking the segments.
Check the file modification times to see which segment contains the earlier data.
If you want to reduce the surplus duration, decrease segment_time and increase segment_wrap correspondingly. segment_time x segment_wrap
should be target saved duration + segment_time
long.
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