In a massive abuse of AWS Lambda, this project is a fully-functional URL shortener built using only lambda - no data store like redis or dynamodb.
To be clear, this is a Bad Idea®. There are many better ways to do this. You should not use this code for stuff. This is very much a "can it be done?" kinda thing.
A full writeup is on kevinkuchta.com
This repo is set up as an Apex project - use that to deploy and test this. There's a lot of configuration around API Gateway that's not really covered in this repo that you'd need to get everything working, though. A live example is or should be up at kmk.party.