The bipolar nebula Menzel 3 (Mz 3) was observed as part of the Herschel Planetary Nebula Survey (HerPlaNS), which used the PACS and SPIRE instruments aboard the Herschel Space Observatory to study a sample of planetary nebulae (PNe). In this paper, one of the series describing HerPlaNS results, we report the detection of HI recombination lines (HRLs) in the spectrum of Mz 3. Inspection of the spectrum reveals the presence of 12 HRLs in the 55680 mu m range covered by the PACS and SPIRE instruments (H11 alpha to H21 alpha and H14 beta). The presence of HRLs in this range is unusual for PNe and has not been reported in Mz 3 before. Our analysis indicates that the HRLs we observed are enhanced by laser effect occurring in the core of Mz 3. Our arguments for this are (i) the available Mz 3 optical to submillimetre HRL alpha line intensity ratios are not well reproduced by the spontaneous emission of optically thin ionized gas, as would be typical for nebular gas in PNe; (ii) the compact core of Mz 3 is responsible for a large fraction of the Herschel HRLs emission; (iii) the line intensity ratios for Mz 3 are very similar to those in the core emission of the well known star MWC 349A, where laser effect is responsible for the enhancement of HRLs in the Herschel wavelength range; (iv) the physical characteristics relevant to cause laser effect in the core of MWC 349A are very similar to those in the core of Mz 3.