Knowledge on the occurrence rate of extreme solar storms is strongly limited by the relatively recent advent of satellite monitoring of the Sun. To extend our perspective of solar storms prior to the satellite era and because atmospheric ionization induced by solar energetic particles (SEPs) can lead to the production of odd nitrogen, nitrate spikes in ice cores have been tentatively used to document both the occurrence and intensity of past SEP events. However, the reliability of the use of nitrate in ice records as a proxy for SEP events is strongly debated. This is partly due to equivocal detection of nitrate spikes in single ice cores and possible alternative sources, such as biomass burning plumes. Here we present new continuous high-resolution measurements of nitrate and of the biomass burning species ammonium and black carbon, from several Antarctic and Greenland ice cores. We investigate periods covering the two largest known SEP events of 775 and 994 Common Era as well as the Carrington event and the hard SEP event of February 1956. We report no coincident nitrate spikes associated with any of these benchmark events. We also demonstrate the low reproducibility of the nitrate signal in multiple ice cores and confirm the significant relationship between biomass burning plumes and nitrate spikes in individual ice cores. In the light of these new data, there is no line of evidence that supports the hypothesis that ice cores preserve or document detectable amounts of nitrate produced by SEPs, even for the most extreme events known to date.