The Ferromagnetic Inductively Coupled Plasma (FICP) source, which is a version of the common inductively coupled plasma sources, has a number of well known advantages such as high efficiency, high level of ionization, low minimal gas pressure, very low required driver frequency, and even a possibility to be driven by single current pulses. We present an experimental study of such an FICP source which showed that above a certain value of the driving pulse power the properties of this device changed rather drastically. Namely, the plasma became non-stationary and non-uniform contrary to the stationary and uniform plasmas typical for this kind of plasma sources. In this case the plasma appeared as a narrow dense spike which was short compared to the driving pulse. The local plasma density could exceed the neutral atoms density by a few orders of magnitude. When that happened, the afterglow plasma decay time after the end of the pulse was long compared to an ordinary case with no plasma spike. Experiments were performed with various gases and in a wide range of pressures which enabled us to understand the physical mechanism and derive the parameters responsible for such plasma behavior. A qualitative model of this phenomenon is discussed.