# [Zsh], <s>19</s> 17 bytes

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    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]: https://www.zsh.org/
[Try it online!]: https://tio.run/nexus/zsh#VYxLC4JAGEXX@Ssuw8T0wHLTJkWIMAjKFrmchTAPFPKb0Ilc9N8tqU13ceCcxTWqchCD1gmP0vJZxqY3CjwaBF7wLUINIUkghbo5MoGqGqex7H/q68a4h8emw2r9Tda1sKgJixjaBZN7W5O3YIfjKct352wradpJCsdJ2l/yIsuL639lYNyOmCXczlmgP8fDGw "Zsh – TIO Nexus"