There's debate around who pioneered the blast beat first. Many cite Napalm Death (UK), but they were heavily influenced by extreme hardcore bands such as Siege (Boston, USA), among others. It would be fair to say that Swedish band Asocial are among the first, with their 1982 EP, How Could Hardcore Be Any Worse?
Other extreme hardcore bands around 1982/83 who used blast beats include Youth Korps (USA):
and DRI (USA):
A little later (1984) Siege was putting extreme hardcore on the map:
Worthy of a mention is Deep Wound (USA) who were pretty fast for 1983 (and feature future Dinosaur Jr members):
Gang Green (USA) gets overlooked but they too were super fast on their first EP:
Total Chaoz (Pre-Larm) from Holland are also worthy of a mention:
I'm sure I've missed a heap of bands (especially Japanese HC and other European HC bands), as well as those outside of hardcore who were using blast beats in the 1960's (Jazz musicians, for example).