Your question is somewhat broad, since local and p-adic fields permeate number theory and other parts of mathematics. If you want to get an idea of which journals publish articles about aspects of p-adic fields, you can look on MathSciNet. For example, there's an entire category

• 11S Algebraic number theory: local and p-adic fields

with 14 subcategories, and there are subcategories in other sections that deal with local fields, including

• 11D88 Diophantine equations: p-adic and power series fields
• 11E95 Forms and linear algebraic groups: p-adic theory
• 11F85 p-adic theory, local fields: Discontinuous groups and automorphic forms
• 11G25 Arithmetic algebraic geometry: Varieties over finite and local fields
• 11K41 Probabilistic theory: distribution modulo 1; metric theory of algorithms: Continuous, p-adic and abstract analogues

So for example, if you're interested in Galois cohomology associated to local fields, you could search on 11S25 in the primary and secondary fields for the year 2021. I did that and found that there are such articles published in Trans AMS, RIMS, IJNT, Proc AMS, Selecta Math, Ann Inst Grenoble, J Inst Math Jusseiu, Mem Soc Math Fr, Acta Math Sin, and Acta Arith. So quite a variety of journals to choose from, and that's just 2021 articles in this particular part of p-adic number theory.

Your question is somewhat broad, since local and p-adic fields permeate number theory and other parts of mathematics. If you want to get an idea of which journals publish articles about aspects of p-adic fields, you can look on MathSciNet. For example, there's an entire category

• 11S Algebraic number theory: local and p-adic fields

with 14 subcategories, and there are subcategories in other sections that deal with local fields, including

• 11D88 Diophantine equations: p-adic and power series fields
• 11E95 Forms and linear algebraic groups: p-adic theory
• 11F85 p-adic theory, local fields: Discontinuous groups and automorphic forms
• 11G25 Arithmetic algebraic geometry: Varieties over finite and local fields
• 11K41 Probabilistic theory: distribution modulo 1; metric theory of algorithms: Continuous, p-adic and abstract analogues

So for example, if you're interested in Galois cohomology associated to local fields, you could search on 11S25 in the primary and secondary fields for the year 2021. I did that and found that there are such articles published in Trans AMS, RIMS, IJNT, Proc AMS, Selecta Math, Ann Inst Fourier, J Inst Math Jussieu, Mem Soc Math Fr, Acta Math Sin, and Acta Arith. So quite a variety of journals to choose from, and that's just 2021 articles in this particular part of $p$-adic number theory.