We present the first [C II] 158 mu m luminosity function (LF) at z similar to 5 from a sample of serendipitous lines detected in the ALMA Large Program to INvestigate [C II] at Early times (ALPINE). A study of the 118 ALPINE pointings revealed several serendipitous lines. Based on their fidelity, we selected 14 lines for the final catalog. According to the redshift of their counterparts, we identified eight out of 14 detections as [C II] lines at z similar to 5, along with two as CO transitions at lower redshifts. The remaining four lines have an elusive identification in the available catalogs and we considered them as [C II] candidates. We used the eight confirmed [C II] and the four [C II] candidates to build one of the first [C II] LFs at z similar to 5. We found that 11 out of these 12 sources have a redshift very similar to that of the ALPINE target in the same pointing, suggesting the presence of overdensities around the targets. Therefore, we split the sample in two (a "clustered" and "field" subsample) according to their redshift separation and built two separate LFs. Our estimates suggest that there could be an evolution of the [C II] LF between z similar to 5 and z similar to 0. By converting the [C II] luminosity to the star-formation rate, we evaluated the cosmic star-formation rate density (SFRD) at z similar to 5. The clustered sample results in a SFRD similar to 10 times higher than previous measurements from UV-selected galaxies. On the other hand, from the field sample (likely representing the average galaxy population), we derived a SFRD similar to 1.6 higher compared to current estimates from UV surveys but compatible within the errors. Because of the large uncertainties, observations of larger samples will be necessary to better constrain the SFRD at z similar to 5. This study represents one of the first efforts aimed at characterizing the demography of [C II] emitters at z similar to 5 using a mm selection of galaxies.