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We present the first [CII] 158 $μ$m luminosity function (LF) at $z\sim 5$ from a sample of serendipitous lines detected in the ALMA Large Program to INvestigate [CII] at Early times (ALPINE). A search performed over 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 8 out of 14 detections as [CII] lines at $z\sim 5$, and two as CO transitions at lower redshifts. The remaining 4 lines have an elusive identification in the available catalogs and we considered them as [CII] candidates. We used the 8 confirmed [CII] and the 4 [CII] candidates to build one of the first [CII] LFs at $z\sim 5$. 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" sub-sample) according to their redshift separation and built two separate LFs. Our estimates suggest that there could be an evolution of the [CII] LF between $z \sim 5$ and $z \sim 0$. By converting the [CII] luminosity to star formation rate we evaluated the cosmic star formation rate density (SFRD) at $z\sim 5$. The clustered sample results in a SFRD $\sim 10$ 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 $\sim 1.6$ higher compared to current estimates from UV surveys but compatible within the errors. Because of the large uncertainties, observations of larger samples are necessary to better constrain the SFRD at $z\sim 5$. This study represents one of the first efforts aimed at characterizing the demography of [CII] emitters at $z\sim 5$ using a mm-selection of galaxies.