Context. Millimetre-wave astronomy is an important tool for both general astrophysics studies and cosmology. A large number of unidentified sources are being detected by the large field-of-view continuum instruments operating on large telescopes. Aims. New smart focal planes are needed to bridge the gap between large bandwidth continuum instruments operating on single dish telescopes and the high spectral and angular resolution interferometers (e.g. ALMA in Chile, NOEMA in France). The aim is to perform low-medium spectral resolution observations and select a lower number of potentially interesting sources, i.e. high-redshift galaxies, for further follow-up. Methods. We have designed, fabricated and tested an innovative on-chip spectrometer sensitive in the 85-110~GHz range. It contains sixteen channels selecting a frequency band of about 0.2 GHz each. A conical horn antenna coupled to a slot in the ground plane collects the radiation and guides it to a mm-wave microstrip transmission line placed on the other side of the mono-crystalline substrate. The mm-wave line is coupled to a filter-bank. Each filter is capacitively coupled to a Lumped Element Kinetic Inductance Detector (LEKID). The microstrip configuration allows to benefit from the high quality, i.e. low losses, mono-crystalline substrate, and at the same time prevents direct, i.e. un-filtered, LEKID illumination. Results. The prototype spectrometer exhibit a spectral resolution R = lambda / Delta_lambda = 300. The optical noise equivalent power is in the low 1E-16W/sqrt(Hz) range for an incoming power of about 0.2pW per channel. The device is polarisation-sensitive, with a cross-polarisation lower than 1% for the best channels.