It is possible to enable this for your AskNicely account - it can be a really good way to get people to be completely honest with their responses, or if you're using your account for an internal Employee NPS survey.
That said, we can't recommend it in good faith. Besides it not being a true representation of the codified NPS approach (and skewing your scores as a result), one of AskNicely's greatest features is its ability to close the loop with your customers and respond directly to their feedback - the good, the bad, and the indifferent. By making surveys anonymous, you lose the ability to make connections with your customers by responding directly to the joyful moments or harsh pain points they have brought your attention. Contact your CSM if you truly believe this is the only way forward for your AskNicely usage.