In my world a data lake is not a data warehouse.
The data lake contains data in as near to the raw source data format as possible at the time it was captured.  The general idea is that it is cheap storage into which data can be stored indefinitely.  It isn't a database as traditionally described.  I'd say that, yes, it should be immutable.

The data warehouse is a thing of rigour and discipline where facts become available at different times and therefore updates may be required.  If I am using the Kimball model then I have the various slow changing dimension (SCD) approaches to choose from.  This has to be driven by business requirements.  As the business requirements change then the data lake means I can revisit the SCD strategy on the basis of those requirements.

Even if I tilt towards something more Inmon esque I may still need the concept of updateable data due to facts becoming available out of sequence.  For example, the process of switching an energy supplier is a lengthy process for many reasons including legislated cooling off periods.  There is valuable insight (subject to GDPR permissions from the customer) even though the data won't be fully complete until 2 months after the original application.