There are possible variants of this question. Let's first ask what is the 'Linux' kernel. Linux is basically some variation on Unix. The Linux kernel provides services as memory management, ...:
The kernel parts and the interfaces all assume some C compiler and C language conventions are used. A C compiler can generate a compact executable.
Now things get difficult in Lisp. It shows in SBCL. Fixnums in SBCL are 61bit and not 64bit. Some bits in data are used for tags (and possibly other things). SBCL programs start at multi-megabyte size. Characters are not bytes, but again tagged data. Most Lisp implementations use tagged data, because the data needs to be identifyable at runtime - in Lisp.
C has very different data structures, etc from Lisp. If you write something in Lisp it would by default look very different from C code. Wouldn't a scheduler be best a CLOS object? Tasks? Timers? But the interfaces of the Linux kernel look very different. You would program very primitive data structures in a language that has higher-level data types (like CLOS). That's not a very good fit.
So, can you define a Lisp that is nearer to C in its capabilities? That's possible. There are some examples.
Also, can you write an OS that would not look like Linux, but is written in Lisp and is more idiomatic? That's also possible and has been done before.
The typical Unix (and Linux) OS 'kernel' does a few things that were not done in the typical (existing) Lisp OS: different applications with their own memory space, multi-user capabilities, ... that would be new ground for a Lisp-based OS and had to be invented first: how to provide multiple independent Lisp applications on top of a Lisp OS and not lose the advantages of a typical Lisp system.
Probably it would be better to not write an OS in C (for example one might want a language and implementation with real array bounds checking) - but the language better be near the C model - otherwise lots of knowhow is not transferrable. Lisp solutions usually look very different.
A real Lisp OS looks very different.