Transient astronomical events that exhibit no discernible association with a host galaxy are commonly referred to as hostless. These rare phenomena can offer unique insights into the properties and evolution of stars and galaxies. However, the sheer number of transients captured by contemporary high-cadence astronomical surveys renders the manual identification of all potential hostless transients impractical. Therefore, creating a systematic identification tool is crucial for studying these elusive events. We present the ExtragaLactic alErt Pipeline for Hostless AstroNomical Transients (ELEPHANT), a framework for filtering hostless transients in astronomical data streams. It was designed to process alerts from the Zwicky Transient Facility (ZTF) presented in the Fink broker; however, its underlying concept can be applied to other data sources. We used Fink to access all the ZTF alerts produced between January 2022 and December 2023, selecting alerts associated with extragalactic transients reported in SIMBAD or TNS, as well as those classified as supernovae (SNe) or kilonovae (KNe) by the machine learning (ML) classifiers within the broker. We then processed the associated stamps using a sequence of image analysis techniques to retrieve hostless candidates. We find that less than 2% of all analyzed transients are potentially hostless. Among them, only around 10 percent have a spectroscopic class reported on TNS, with type Ia SNe being the most common class, followed by superluminous SNe. In particular, among the hostless candidates retrieved by our pipeline, there is SN 2018ibb, which has been proposed to be a pair instability SN candidate, and SN 2022ann, one of only five known SNe Icn. When no class is reported on TNS, the dominant classes are quasi-stellar object (QSO) and SN candidates, with the former obtained from SIMBAD and the latter inferred using the Fink ML classifier. ELEPHANT represents an effective strategy to filter extragalactic events within large and complex astronomical alert streams. There are many applications for which this pipeline will be useful, ranging from transient selection for follow-up to studies of transient environments. The results presented here demonstrate the feasibility of developing specially crafted pipelines that enable a variety of scientific studies based on large-scale surveys.