Has anyone experimented with serverless applications in Kotlin (e.g. with knative or AWS Lambda)? If so, with what "flavour" of Kotlin (JVM, JS, Native) ? And is the performance acceptable for the serverless world (especially for Kotlin JVM that has worse warm up time) ?

Kotlin Native (currently in Beta) is one option. In fact there is a official Serverless Kotlin Native sample: https://github.com/JetBrains/kotlin-native/tree/master/samples/weather_function

I wouldn't recommend the Java/JVM support of AWS Lambda. If you have some library dependencies, class loading can take many seconds. I learned this the hard way when I built an Alexa skill using Kotlin/JVM. One workaround is to use Graal to build a native executable: https://engineering.opsgenie.com/run-native-java-using-graalvm-in-aws-lambda-with-golang-ba86e27930bf

Unfortunately Kotlin Native and GraalVM are quite new and not really battle tested yet, right?

Yes, that's true.

I also experienced the slow startup when we developed an Alexa skill with Kotlin/JVM. That said, you can of course employ strategies to keep your lambdas "warm" to avoid the long startup times.

Kotlin Native and GraalVM are currently at the exploration ⛰️ phase 🙃.