Yes, take a look at AWS Greengrass: https://aws.amazon.com/greengrass/

This lets you run the same Lambda execution environment on any device you own. Typically, we see this being used by customers that need to do computing at the edge where Internet connectivity is poor or intermittent (think oil and gas fields, mines, etc.)

Disclaimer: I work for AWS

No, but take a look at OpenFaaS or FaaS-netes, similar concept but running on-premises. Definitely less mature, but they can get you 90% of the way there if you're not doing anything super complicated.

https://github.com/openfaas/faas (Serverless for on-prem, roll your own infra, etc)

What's the benefit or use case of running serverless on-premise? Isn't the whole ideal of serverless is that you do not maintain servers yourself? Instead of specific Lambda/FaaS solution for on-prem, could we not just use docker package the function in startup script to achieve the same?

I don't believe it would, AWS Lambda is all hosted at the moment. We moved some on-prem systems into EC2 and finally to Lambda functions.