Branches of foreign banks are not permitted. Foreign investment in the banking sector is only allowed via a local incorporated institution.
"Article 2 (Definitions of terms) 
Terms used in this Law shall have the following meanings:
a) A bank shall be understood to mean a joint stock company headquartered in the Federation of Bosnia and Herzegovina (hereinafter: the Federation), which is licensed by the Banking Agency of the Federation of Bosnia and Herzegovina (hereinafter: the Agency), performing the activity of taking deposits and other repayable funds and lending for its own account, and which may carry out other transactions in conformity with this Law. 
b) A foreign bank is a legal entity headquartered abroad which has an operating licence of the regulatory body of the home country, established as a bank, and registered with the competent authority of that country.   
c) Organisational units of banks and banks seated in Republika Srpska (hereinafter: RS) or the Brčko District of Bosnia and Herzegovina (hereinafter: the Brčko District) shall constitute branches, as basic business units, and lower-level organisational units subordinated to the branch, such as sub-branches, outposts, agencies and the like, which do not have legal personality and carry out all or part of the operations that may be performed by a bank in accordance with this Law. 
d) A representative office in terms of this Law shall be understood to mean an organisational unit of a bank opened abroad or of a bank outside the Federation, opened in the Federation, without legal personality, which may not perform banking activities, and which only performs activities of market research and providing information, as well as presenting and promoting the bank that established it. 
[...]
Article 7 (Legal form and founders)
(1) Banks are business companies established in the legal form of joint stock companies. 
(2) Banks may be established by national and/or foreign legal or natural persons. 
(3) Banks shall be considered to be open joint stock companies, regardless of the number of shareholders. [...]" (Law on Banks of FB&H (Official Gazette of FB&H No.27/17)
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"Meaning of Individual Terms
Article 2  
The terms used in this Law shall have the following meaning:  
1) A bank is a joint-stock company with a head office in Republika Srpska, which has an operating license, issued by the Banking Agency of Republika Srpska (hereinafter: Agency), whose activity is to receive deposits or other repayable funds and to provide loans for its own account, but may also conduct other operations in accordance with this Law,  
 2) A foreign bank is a legal person with a head office abroad which has an operating license issued by the regulatory authority of the country of origin and which has been established as a bank and is entered into the register of the competent authority of that country,   
3) Organizational parts of a bank and banks with a head office in the Federation of Bosnia and Herzegovina or the Brčko District of Bosnia and Herzegovina are affiliates operating as basic business units, and lower organizational parts dependent on the affiliates, such as branches, counters/desks, agencies and similar, which are not in the capacity of a legal person, but conduct all operations or a portion of operations that can be conducted by a bank in accordance with this Law,   
4) A representative office is an organizational part of a bank established abroad or a bank outside of Republika Srpska established in Republika Srpska, without the capacity of a legal person, which cannot conduct the bank operations, and which conducts only operations related to market research and provision of information, as well as representation and promotion of the bank that has established it,
[...]
CHAPTER II ESTABLISHMENT AND CLOSING OF A BANK 
1.  Establishment of a Bank
Founders and legal form 
Article 11 
(1) A bank is a company that is established in the legal form of a joint-stock company.  
(2) A bank may be established by domestic and/or foreign legal and/or natural persons.   
(3) A bank is considered as an open joint-stock company regardless of the number of shareholders.  [...]" (Law on Banks of RS (Official Gazette of RS No. 3/17 and 54/19)

Law on Banks of FB&H (Official Gazette of FB&H No.27/17)
Law on Banks of RS (Official Gazette of RS No. 3/17 and 54/19)

[<a href="https://www.fba.ba/images/zakoni_banke/zakon_o_bankama_fbih_eng.pdf">Law</a>],[<a href="https://abrs.ba/data/documents/208/RS_Banking_Law_April_2017_Revision_2021.pdf">Law</a>],[<a href="http://fipa.gov.ba/publikacije_materijali/brosure/Starting%20and%20operating%20business%20in%20BiH%202022.pdf">Other</a>]