How to Not allow the PUT method at all but allow PATCH in a DRF ViewSet?

Instead of using mixins.UpdateModelMixin just define your own mixin that would perform patch only:

class UpdateModelMixin(object):
    """
    Update a model instance.
    """
    def partial_update(self, request, *args, **kwargs):
        partial = True
        instance = self.get_object()
        serializer = self.get_serializer(instance, data=request.data, partial=partial)
        serializer.is_valid(raise_exception=True)
        self.perform_update(serializer)

        if getattr(instance, '_prefetched_objects_cache', None):
            # If 'prefetch_related' has been applied to a queryset, we need to
            # forcibly invalidate the prefetch cache on the instance.
            instance._prefetched_objects_cache = {}

        return Response(serializer.data)

    def perform_update(self, serializer):
        serializer.save()

If you want to use builtin mixins.UpdateModelMixin, limit to PATCH and disable swagger from showing PUT you can use http_method_names

class UserViewSet(mixins.UpdateModelMixin, GenericViewSet):
    queryset = User.objects.all()
    serializer_class = UserSerializer
    http_method_names = ["patch"]

A simple and straight forward approach:

class UserViewSet(viewsets.ModelViewSet):
    queryset = User.objects.all()
    serializer_class = UserSerializer
    http_method_names = ['get', 'post', 'patch'] # <---------

Like this the PUT method will not be allowed.

I think a superior solution would be to use a custom router and disable the route for PUT. Then use your custom router for the viewsets.

class SimpleRouter(BaseRouter):
    routes = [
        # List route.
        Route(
            url=r'^{prefix}{trailing_slash}$',
            mapping={
                'get': 'list',
                'post': 'create'
            },
            name='{basename}-list',
            initkwargs={'suffix': 'List'}
        ),
        # Dynamically generated list routes.
        # Generated using @list_route decorator
        # on methods of the viewset.
        DynamicListRoute(
            url=r'^{prefix}/{methodname}{trailing_slash}$',
            name='{basename}-{methodnamehyphen}',
            initkwargs={}
        ),
        # Detail route.
        Route(
            url=r'^{prefix}/{lookup}{trailing_slash}$',
            mapping={
                'get': 'retrieve',
                'put': 'update',
                'patch': 'partial_update',
                'delete': 'destroy'
            },
            name='{basename}-detail',
            initkwargs={'suffix': 'Instance'}
        ),
        # Dynamically generated detail routes.
        # Generated using @detail_route decorator on methods of the viewset.
        DynamicDetailRoute(
            url=r'^{prefix}/{lookup}/{methodname}{trailing_slash}$',
            name='{basename}-{methodnamehyphen}',
            initkwargs={}
        ),
    ]

^ The router implementation looks something like that. So you just need to inherit the SimpleRouter, or perhaps the DefaultRouter, and defines the routes class attribute how you want it. You can remove the mapping for 'put' in the Route(mapping={...}) completely, or you can define your own action to handle it and return the appropriate 400-something resonse.