I have had this happen with two of these lathes. It is not supposed to do that & there is nothing that is supposed to hold it in place from the ends. It is designed to be machine fit into the bearings, basically fit so clean & tight that the metals lock together. However, it does not work for very long. The heat generated by running the lathe expands the spindle metal more than the hardened bearing metal, and then it cools off & expands again when ran again. After some time it'll just release & begin to free-ball. Drilling accentuates the issue. What you can do to lock it back up is use some loc-tite around the seams. It'll secure the spindle to the bearings forever. When/if the bearings need replacement simply heat the loc-tite up with a propane torch & it'll set free for easy change.
If you are not comfortable using the loc-tite, then you can close the chuck completely & use a hammer to tap the front of the closed jaws. This will force the entire spindle backwards into the bearings to a point where the spindle locks into the bearings again. It'll be a very tiny bit, so there's no need to pound it. Just tap it until it no longer has that linear movement. I prefer the loc-tite but each method has & does work. No need to tear your machine down for a rebuild. The machine I have now aquired this issue 3 years ago. I used loc-tite & have done nothing else to it, and it gets 10+hrs/week actual running time.