If I'm not putting out a fire, responding to a request, or working on a deliverable, then I'm learning something. I believe that most of us have more opportunity and time for learning (not necessarily formal training) than we realize. This could be taking a deep dive into Google researching a technical topic, watching a video, or working on a proof of concept. If you find that you're swamped all the time, then maybe reading a book on using PowerShell script to automate common DBA tasks would be a good place to start, and then once you start reaping the benefits of that, then use the free time to move on to other things.
"Do not seek to follow in the footsteps of the wise. Instead, seek what they sought." - Matsuo Basho