The first branch of a non-EU bank must be approved by the European Central Bank on the proposal of the Bank of Italy in coordination with the Ministry of Foreign Affairs and subject to reciprocity. Non-EU banks already operating a branch in Italy can open other branches with the authorisation of the Bank of Italy. 
"Art. 14  - Authorization for banking activities 
1. ((Banking authorization is issued)) when the following conditions are met: 
a) the form of joint-stock company or company is adopted cooperative for limited responsabilita actions; 
a-bis) the registered office and general management are located in territory of the Republic; 
b) the paid-up capital is not less than that determined by the Bank of Italy; 
c) a program concerning the activity is presented initial, together with the articles of association and the statute; [...]
e) persons who perform administrative and management functions and control are suitable, pursuant to article 26;  [...]
2. ((The authorization is issued by the ECB, on the proposal of the Bank of Italy; is denied, by the Bank of Italy or the ECB,)) when the verification of the conditions indicated in paragraph 1 does not result sound and prudent management guaranteed.  [...]
4. The establishment in Italy of the first branch of a bank non-EU and 'authorized by the Bank of Italy, having heard the Ministry of Foreign Affairs, subject to compliance with conditions corresponding to those of paragraph 1, letters b), c) and e).
The authorization is issued also taking into account the condition of reciprocity. 
[...]
 Art. 15 Branches 
((01. Italian banks can establish branches in the territory of the Republic and of the other EU states in compliance with the procedures envisaged by the provisions of the SSM. The banks of other EU countries may establish branches in territory of the Republic in accordance with the established procedures by the provisions of the SSM and, for the banks of the EU member states not participating in the SSM, of paragraph 3.))  [...]
3. ((For banks of EU countries not participating in the SSM who intend to establish branches in the territory of the Republic, the first settlement is preceded by a communication to the Bank of Italy by the competent authority of the State of membership; the branch begins operations two months after communication.))  [...]
4. Non-EU banks already operating in the territory of Repubblica with a branch may establish other branches via authorization from the Bank of Italy.[...]" (LEGISLATIVE DECREE 1 September 1993, n. 385, "Consolidated text of banking and credit laws"; last amended by LAW of 23 July 2021, n. 106; Articles 14 and 15 [google translation])
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"Article 29-bis - Banks of the European Union
1. The banks of the European Union may provide investment services or activities, with or without additional services, in the exercise of the right of establishment through branches or associate agents established in Italy. The establishment of branches is governed by article 15 of the consolidated banking law. The provision of investment services through associate agents is preceded by a communication to CONSOB, in conformance with the technical regulation and implementation regulations indicated in article 27, paragraph 1.
[...]
Article 29-ter - Banks of non-EU countries
1. In the case where the performance of investment services or activities, with or without additional services is envisaged, the establishment of branches by banks by banks of non-EU countries in Italy is authorised by the Bank of Italy, having consulted CONSOB, when the conditions referred to in article 28, paragraph 1 obtain. This is without prejudice to the application of the articles 13, 14, paragraph 4, and 15, paragraph 4, of the consolidated banking law.
2. The Authorisation referred to in paragraph 1 is denied if the capacity of the branch of the bank of the non-EU country to respect the obligations applicable to it pursuant to this decree or contained in directly applicable deeds of the European Union is not guaranteed. [...]"(LEGISLATIVE DECREE of February 24, 1998, n. 58, "Consolidated text of the provisions on financial intermediation, pursuant to articles 8 and 21 of law no. 52"; last amended by LEGISLATIVE DECREE 2 February 2021, n. 17; Articles 29-bis and 29-ter [unofficial translation])
LEGISLATIVE DECREE 1 September 1993, n. 385, "Consolidated text of banking and credit laws"; last amended by LAW of 23 July 2021, n. 106; Articles 14 and 15
LEGISLATIVE DECREE of February 24, 1998, n. 58, "Consolidated text of the provisions on financial intermediation, pursuant to articles 8 and 21 of law no. 52"; last amended by LEGISLATIVE DECREE 2 February 2021, n. 17; Articles 29-bis and 29-ter 
[<a href="http://www.normattiva.it/uri-res/N2Ls?urn:nir:stato:decreto.legislativo:1993-09-01;385!vig=2021-08-20">Decree</a>],[<a href="http://www.normattiva.it/uri-res/N2Ls?urn:nir:stato:decreto.legislativo:1998-02-24;58!vig=2021-08-20">Decree</a>],[<a href="https://www.consob.it/web/consob-and-its-activities/laws-and-regulations/documenti/english/laws/fr_decree58_1998.htm">Decree_Eng</a>]