Is it possible to distribute the code of a class to several files?

Honestly, I think the best route is to break it up into different roles that you compose into your class.

After all, how are you planning on breaking up your class?
Are you going to group methods and attributes according to similarity?

At that point you've just about come up with a role, so you might as well make it a role.

If you look at the source for Rakudo you see things like this:

class Perl6::Metamodel::ClassHOW
    does Perl6::Metamodel::Naming
    does Perl6::Metamodel::Documenting
    does Perl6::Metamodel::LanguageRevision
    does Perl6::Metamodel::Stashing
    does Perl6::Metamodel::AttributeContainer
    does Perl6::Metamodel::MethodContainer
    does Perl6::Metamodel::PrivateMethodContainer
    does Perl6::Metamodel::MultiMethodContainer
    does Perl6::Metamodel::MetaMethodContainer
    does Perl6::Metamodel::RoleContainer
    does Perl6::Metamodel::MultipleInheritance
    does Perl6::Metamodel::DefaultParent
    does Perl6::Metamodel::C3MRO
    does Perl6::Metamodel::MROBasedMethodDispatch
    does Perl6::Metamodel::MROBasedTypeChecking
    does Perl6::Metamodel::Trusting
    does Perl6::Metamodel::BUILDPLAN
    does Perl6::Metamodel::Mixins
    does Perl6::Metamodel::ArrayType
    does Perl6::Metamodel::BoolificationProtocol
    does Perl6::Metamodel::REPRComposeProtocol
    does Perl6::Metamodel::InvocationProtocol
    does Perl6::Metamodel::ContainerSpecProtocol
    does Perl6::Metamodel::Finalization
    does Perl6::Metamodel::Concretization
    does Perl6::Metamodel::ConcretizationCache
{
    … # only 300 lines of code
}

If you do a good job of splitting up your roles, you should be able to use them in many classes.
How many classes in the Rakudo code base do you think compose in the Perl6::Metamodel::Naming role?

That role only provides a few things, and is only 45 lines long.

Here is an abbreviated version. (All of the code in the methods has been deleted here for brevity.)

role Perl6::Metamodel::Naming {
    has $!name;
    has $!shortname;

    method name($obj) {
        …
    }
    method set_name($obj, $name) {
        …
    }
    method shortname($obj) {
        …
    }
    method set_shortname($obj, $shortname) {
        …
    }
}

Yes, there always is. But there us no standard supported way (yet anyway).

1. You can take the approach that Raku takes itself when creating the core settings: concatenate the files into a single file, and compile that. When you're building Rakudo from scratch, that's what's happening when you see the line:

   +++ Generating gen/moar/foo
   

   The generated files can be inspected in the gen/moar directory. At one point, I brought this up in a problem solving issue, but that never went anywhere. Perhaps that should be revisited.

2. You can use augment class. But that currently only makes sense inside a single file, because with multiple files it is creating multiple versions of the same "module" in pre-compilation. And the system is not able to determine what to resolve to what. This is when you realize that pre-compilation of Raku modules currently is creating libraries that need to be statically linkable at runtime.