German officials have said that the country's intelligence service paid an informant more than $6.2m for a CD believed to contain 1,400 names of alleged tax cheats, about 600 of them Germans.
The information obtained by German intelligence has led to tax evasion investigations in Germany, France, Britain, Australia, Italy, New Zealand, Sweden, the United States and elsewhere.
Liechtenstein's LGT Group said last month that it believed the data was stolen from its subsidiary LGT Treuhand by Kieber, a former employee.
The bank is wholly owned by the principality's ruling family.
The Liechtenstein police notice refers to media reports that Kieber was issued with a new identity and travel documents by German intelligence.
It says Liechtenstein will immediately demand Kieber's extradition if he is arrested abroad.
Liechtenstein has protested against the methods used by Germany to acquire the customer information.
Europe's fourth-smallest sovereign state has become wealthy through its status as an offshore tax haven, with strict rules on banking secrecy.
The principality insists that it will hold onto its practice of allowing foreigners to open trusts anonymously by registering them through a local attorney or trustee.
The Paris-based organisation for economic co-operation and development, which advises on good practice in financial affairs, has put Liechtenstein on a blacklist of unco-operative tax havens.