Learning is itself a skill worth focusing on. I learn new tools, languages, and concepts all the time even though I often don't do anything with them.
I think that fear plays a lot into our ability to learn new things. Fear of change. Fear of failure. Fear of starting over.
But things are always going to be changing, it's worthwhile to figure out how you can best adapt and keep moving. It's not reasonable to think that you're aren't going to have to learn new things throughout your career, developer or not.
At the same time, don't worry too much about keeping up with the latest thing. Spread yourself out a bit and learn lots of very different topics. They tend to reinforce one another.