/ AbstractNenuFAR (New extension in Nançay upgrading LoFAR) is a new radio telescope developed and built on the site of the Nançay Radio Observatory. It is designed to observe the largely unexplored frequency window from 10 to 85\,MHz, offering a high sensitivity across its full bandwidth. NenuFAR has started its "early science" operation in July 2019, with 58\% of its final collecting area being available. Pulsars are one of the major topics for the scientific exploitation of this frequency range and represent an important challenge in terms of instrumentation. Designing instrumentation at these frequencies is complicated by the need to compensate for the effects of both the interstellar medium and the ionosphere on the observed signal. Our real-time pipeline LUPPI (Low frequency Ultimate Pulsar Processing Instrumentation) is able to cope with a high data rate and to provide real-time coherent de-dispersion down to the lowest frequencies reached by NenuFAR (10\,MHz). The full backend functionality is described, as well as the main pulsar observing modes (folded, single-pulse, waveform, and dynamic spectrum). This instrumentation allowed us to detect 172 pulsars in our first targeted search below 85\,MHz, including 10 millisecond pulsars (6 of which detected for the first time below 100 MHz). We also present some of the "early science" results of NenuFAR on pulsars: a high frequency resolution mapping of PSR B1919$+$21's emission profile and a detailed observation of single-pulse sub-structures from PSR~B0809$+$74 down to 16\,MHz, the high rate of giant-pulse emission from the Crab pulsar detected at 68.7\,MHz (43 events/min), and the illustration of the very good timing performance of the instrumentation, allowing us to study dispersion measure variations in great detail.