Political asylum, usually called simply asylum, was a diplomatic procedure by which a sovereign entity granted someone protection from a hostile force.
In 2153, when the cogenitor "Charles" expressed a desire to leave the Vissian culture and remain aboard the Enterprise NX-01, Captain Jonathan Archer considered it to be a request for asylum and informed the Vissian captain Drennik that it was something he had to take seriously. Ultimately, however, Archer chose not to grant this asylum and "Charles" later committed suicide. (ENT: "Cogenitor")
In 2366, when Admiral Alidar Jarok's Romulan scout ship was pursued by a Romulan warbird and he reached Federation space and requested asylum aboard the USS Enterprise-D. (TNG: "The Defector")
In 2367, the Kaelon Timicin requested asylum aboard the USS Enterprise-D in an attempt to challenge the ritualistic suicide custom called The Resolution. The same year, Klingon Ambassador Kell requested asylum aboard the USS Enterprise-D after he was exposed as an operative working for the Romulans. Captain Jean-Luc Picard, however, refused to grant asylum unless he was found innocent of any involvement in the attempted assassination of Governor Vagh. (TNG: "Half a Life", "The Mind's Eye")
In 2368, Commander William T. Riker asked the J'naii leader Noor to allow Soren asylum on board the USS Enterprise-D in lieu of psychotectic treatment. (TNG: "The Outcast")
In 2369, the terrorist Tahna Los requested asylum from the Federation against the Cardassians. The same year, Miles O'Brien encouraged Tosk to request asylum, Tosk instead chose to die with honor. (DS9: "Past Prologue", "Captive Pursuit")
In 2372, while in the Delta Quadrant, the USS Voyager encountered a Q who would later be known as Quinn, who requested asylum from Captain Janeway against the Q Continuum. Because the request was challenged by Q, Janeway convened and presided over a formal hearing, with Tuvok advocating for Quinn. Although it was unclear how exactly Janeway could protect him from the other Q, she granted Quinn's request. (VOY: "Death Wish")