Is it permissible to deposit money in Islamic banks for savings, and what are the names of some reliable Islamic banks in Alexandria, Egypt?

For dealing with Islamic banks to be permissible, their transactions must be governed by Sharia principles. In that case, there is no harm in depositing money in them for savings or investment.

As for Faysal Islamic Bank, it generally adheres to Islamic principles, although occasionally there are some Sharia violations due to the Sharia supervisory board's selection of some weak jurisprudential opinions or the negligence of employees.