Zsh, 19 17 9 bytes

#!/bin/zsh
<$0>$$;$0

Per consensus on meta, the shebang is excluded from the byte count.

Try it online!

Zsh, 19 17 9 bytes

#!/bin/zsh
<$0>$$;$0

Per consensus on meta, the shebang is excluded from the byte count.

Try it online!

Note that TIO's forkbomb protection will kill the process after 113 files have been generated. It can easily generate 1000 files on a system without such conservative resource limits.

Zsh, 19 17 bytes

dd<$0>`w`;exec $0

I think this fits the bill. It might overwrite the same file a few times, but it will continuously generate new ones.

Try it online!

Zsh, 19 17 9 bytes

#!/bin/zsh
<$0>$$;$0

Per consensus on meta, the shebang is excluded from the byte count.

Try it online!

Zsh, 19 bytes

cp $0 "`w`"
exec $0

I think this fits the bill. It might overwrite the same file a few times, but it will continuously generate new ones.

Try it online! (limited to 5 seconds before showing output)

Zsh, 19 17 bytes

dd<$0>`w`;exec $0

I think this fits the bill. It might overwrite the same file a few times, but it will continuously generate new ones.

Try it online!