yes emdr can cause a person to remember memories even small bits of memories. And yes you can work the emdr process with any memory even those that are not complete memories. In fact many people enter emdr with only a small piece of a memory that is bothering them. then through the process more and more of the memory comes to the surface. what matters is the outcome not the memory itself. you stated you felt calm and at peace, well thats what emdr is for, getting rid of the anxiety and stress a person feels about their self, their memories.

I recently did emdr with only one image that was especially annoying and stressful to me.... a chair. thats it a memory of a chair. through the process we focused on the chair.. what it looked like, what I felt when seeing it in my mind, whether it was by a wall, .... eventually through the process this bit memory (or as you call it a fragment of a memory) became clearer as to how and why I was seeing this image of a chair in my mind. now I am no longer bothered by the chair. I am calm and at peace with it and the memory.

my suggestion is talk with your treatment provider that you did the emdr with. they will help you to decide what you want to do with this new little piece of memory and whether in your situation if emdr will work for you like it did for me.