I can’t name specific solutions off the top of my head as I haven’t done it in a while, but Yes. There are many live distros that offer an optional ‘overlayFS’ (I think that’s the right term) which reserves some of your portable drive’s storage for persistent changes to apply over the base live FS.
Combine that with git and/or NextCloud or an SSH file mount or three and you could probably come up with a complete mobile setup that also has quick access to whatever home server infrastructure you need. 128GB or 256GB USB sticks are pretty cheap now which is plenty for a spacious install. Or even a small USB SSD.