Abstract : Context. Pure gas-phase chemistry models do not succeed in reproducing the measured abundances of small hydrocarbons in the interstellar medium. Information on key gas-phase progenitors of these molecules sheds light on this problem. Aims: We aim to constrain the chemical content of the Horsehead mane with a millimeter unbiased line survey at two positions, namely the photo-dissociation region (PDR) and the nearby shielded core. This project revealed a consistent set of eight unidentified lines toward the PDR position. We associate them to the l-C3H+ hydrocarbon cation, which enables us to constrain the chemistry of small hydrocarbons. We observed the lowest detectable J line in the millimeter domain along a cut toward the illuminating direction to constrain the spatial distribution of the l-C3H+ emission perpendicular to the photo-dissociation front. Methods: We simultaneously fit 1) the rotational and centrifugal distortion constants of a linear rotor; and 2) the Gaussian line shapes located at the eight predicted frequencies. A rotational diagram is then used to infer the excitation temperature and the column density. We finally compare the abundance to the results of the Meudon PDR photochemical model. Results: Six out of the eight unidentified lines observable in the millimeter bands are detected with a signal-to-noise ratio from 6 to 19 toward the Horsehead PDR, while the two last ones are tentatively detected. Mostly noise appears at the same frequency toward the dense core, located less than 40'' away. Moreover, the spatial distribution of the species integrated emission has a shape similar to radical species such as HCO, and small hydrocarbons such as C2H, which show enhanced abundances toward the PDR. The observed lines can be accurately fitted with a linear rotor model, implying a 1Σ ground electronic state. The deduced rotational constant value is B = 11 244.9512 ± 0.0015 MHz, close to that of l-C3H. Conclusions: This is the first detection of the l-C3H+ hydrocarbon in the interstellar medium. Laboratory spectroscopy is underway to confirm these results. Interferometric imaging is needed to firmly constrain the small hydrocarbon chemistry in the Horsehead. Based on observations obtained with the IRAM-30 m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).