I had a very good friend who was just like this. We became friends when we were both single, worked together, and had most of our social outings together. It was a very intense friendship, with some of that new friendship “crush” energy. Lots of long, intense talks, heart to hearts, and soul baring. In lots of ways it was a very positive, fun time, in which I think we helped each other through a lot.
Life changed, we both got different jobs, both became less interested in going out. We stayed in close contact, but the demands for connection became more and more demanding. I needed to call several times a week for long conversations, or risk upsetting her, even if I had nothing to say or different and equally important moments in my life that were demanding a lot of my attention.
In all honesty, it felt as if my friend wanted the emotional intensity of a new romantic relationship, but from a friend. Realising that I hadn’t called her for a few days made me feel incredibly stressed, rather than experiencing the warm anticipation of speaking to someone with whom you have a happy, healthy relationship.
I really sympathise, and I don’t have any solutions. My friend struggled to spend time by herself (even when in relationships), and I think that she found it very difficult and took it very personally when people had to prioritise other things in their lives above the friendship. We talked a lot about it and her insecurities, but this was ultimately something that she had to fix for herself. I think the hardest thing was for her to really internalise the truth (that I was busy but nevertheless still deeply cared for her) rather than her fears (that I was snubbing her and no longer cared for her), and she needed the constant contact to feel reassured.
If you have an otherwise good friendship then maybe you could talk to your friend and try to discover what emotions, negative thoughts or fears she’s experiencing when she doesn’t hear from you. Perhaps there’s a way for her to gradually come to see that your friendship is strong, and that her fears are unfounded.