We report a new population of objects discovered using the data from the James Webb Space Telescope, which are characterized by their point-like morphology and narrow permitted emission lines. Our sample includes nine objects found in three JWST wide survey fields, which have z=3.624 to 5.378 and M_B ~ -18.3 to -21.4 mag. Their light distributions follow gaussian profiles, with the full-width-at-half-maximum (FWHM) values only 3.7%--34.6% larger than those of the point spread functions. Their sharpest FWHM sizes correspond to only 0.49 to 0.96 kpc. They have very strong [O III] and Halpha lines (median rest-frame equivalent widths of 1610.6 and 1374.0 A, respectively), and the line widths of the latter are only 150--360 km/s. Due to the limitation of the current data, the exact nature of this new population is still uncertain. The emission line diagnostics shows that at least one object is consistent with being AGN. On the other hand, the spectral energy distributions of most of the nine objects can be fitted reasonably by normal galaxy templates, which suggest that they could also be very young (median age of 120 Myrs), star-forming (median star formation rate of 1.8 Msun/yr) galaxies in the early formation stage (having acquired a median stellar mass of only 10^{8.4} Msun). If they are indeed star-forming galaxies, their gas-phase metallicities range from 12+log(O/H) = 7.8 to 8.3. It will be critical to understand this population by deeper medium-resolution spectroscopy in the future. If they are AGNs, they constitute a new kind of type 2 AGNs that are low-luminosity and almost "hostless". If they are star-forming galaxies, they also constitute a new kind whose early formation is likely a secular process starting from a very compact core, which is in line with the picture of a monolithic collapse.