Not going to kill her but there is an underlying reason for it and it very well can kill the palm tree
Boredom. Just like a horse chews wood when bored, she might be doing it for entertainment.
Pain. There might be a soothing element. (Ulcers)
Hunger. Provide enough hay to where she still has something to pick at hours later. You can offer her free choice hay but a high fiber hay will make her eat more than her fill looking to satisfy a need.
Nutritional deficiency. You do not mention salt and minerals. I'm not talking "mineral" salt blocks as those are deficient in any value as a mineral source. Something you provide loose on the side (purina 12:12, ADM Grostrong, Progressive Nutrition... A quality product with no more than 25% salt, a high Ca value (12%+) and 100,000 IU vit A as a bonus are the keys I look for. I won't pay for a mineral product that is 75% or more salt because the remaining percentage can't be properly balanced for a horse's needs. It will only satisfy their salt needs.
Grains. That can be a + or a - depending upon what and how much she eats. Circling back to ulcers and pain.