A country can choose to extradite a person, even if there is no extradition treaty. If it’s politically convenient, the country may choose to do so. And the US has ways to make it politically convenient: arms deals, foreign aid agreements, and so on. Going to a country without an extradition treaty just means that you’re gambling that the government of that country will never be persuaded to give you up.
Polanski is different. He fled to France, which has a long-standing policy of not extraditing its own citizens. France offered to try him in the French courts, applying US law, but the US declined.
However, when he was arrested in Zurich, the Swiss courts refused extradition, on the basis that they were not satisfied with the documents presented by the US on the key issue of the significance of the pre-trial detention of Polanski in California, as summarised in Wikipedia:
In a press conference held by Swiss Justice Minister Eveline Widmer-Schlumpf, she stated that Polanski’s extradition to the U.S. was rejected, in part, because U.S. officials failed to produce certain documents, specifically “confidential testimony from a January 2010 hearing on Mr. Polanski’s original sentencing agreement”. According to Swiss officials, the records were required to determine if Polanski’s 42-day court-ordered psychiatric evaluation at Chino State Prison constituted Polanski’s whole sentence according to the now-deceased Judge Rittenband. They reasoned that if this was the correct understanding, then “Roman Polanski would actually have already served his sentence and therefore both the proceedings on which the U.S. extradition request is founded and the request itself would have no foundation.”
A similar issue arose in Poland, and the Polish courts ruled that too much time had passed for extradition:
On November 27, 2015, Poland decided it will not extradite Polanski to the U.S. after prosecutors declined to challenge the court’s ruling, agreeing that Polanski had served his punishment and did not need to face a U.S. court again. Preparations for a movie he was working on had been stalled by the extradition request from last year.[126]
On December 6, 2016, the Supreme Court of Poland ruled to reject an appeal filed by Polish Minister of Justice Ziobro, and to uphold the October 2015 ruling.[127]