We study the intensity oscillations in the upper chromosphere/transition region (TR) and corona, above a bright point (BP) in the solar atmosphere. We analyse the time series of He ii 256 Å, Fe xii 195 Å and Fe xv 284 Å, observed in a 40‐arcsec slot close to the centre of the Sun above the BP by the extreme ultraviolet (EUV) imaging spectrometer (EIS) on board Hinode. Using standard wavelet and periodogram tools, we produce power spectra of intensity oscillations. In the He ii 256.32 Å and Fe xii 195.12 Å EUV light curves, we detect intensity oscillations of the periods ∼ 263 ± 80 s and ∼ 241 ± 60 s, respectively, with a probability >95 per cent in wavelets, which are also consistent with their periodograms. This provides the most likely signature of the propagation of acoustic oscillations around the ∼5.0‐min period from the photosphere to the inner corona. The radiative cooling and thus the finite radiative relaxation time are found to be the most likely mechanisms for the reduced cut‐off frequency environment above the observed BP. This may allow the transfer of ∼5.0‐min acoustic oscillations from the upper chromosphere/TR into the corona. We find that intensity oscillations in He ii 256.32 Å show temporal damping during the total span of the observation. This may be the first most likely observational signature of acoustic wave damping in the upper chromosphere caused by the radiative cooling effect. The intensity oscillations in Fe xii 195.12 Å show an amplification, which may be a most likely signature of the mode‐coupling (two‐wave interaction) and then resonant energy conversion, probably from transverse magnetohydrodynamic (MHD) waves of the double period (e.g. Alfvén waves) to the observed acoustic waves in the lower solar atmosphere where the plasma beta tends to unity. However, we find no evidence of real oscillations around the ∼5.0‐min period with its amplification in the higher corona where the Fe xv 284.16 Å line is formed, which rules out this type of wave activity there. Almost 1.6 per cent of the solar surface is covered with small BPs, probably associated with the small‐scale closed‐loop system, which may be a subset of expanding flux tubes. Hence, the leakage of ∼5.0‐min oscillations above such BPs, which is associated with the highest powers of strong convective motions, and probably resonantly amplified by transverse MHD waves (e.g. Alfvén waves), may be significant for heating the solar atmosphere locally.