Can I open an account with Zerodha if I have an existing NRE-PIS account with ICICI Bank?

Can he open the trading account with Zerodha without opening any other account with HDFC, Axis or Yes bank?

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Yes, you can open multiple demat accounts with other brokers.

No, you cannot open an account at Zerodha by mapping your NRE-PIS account with ICICI Bank.
Zerodha hasn’t yet started offering PIS services through ICICI. You will have to come through either one of these three banks banks only - HDFC, Axis or Yes bank. These are the 3 banks with which Zerodha offers PIS services.

An NRI cannot have multiple Demat accounts. You can have one NRE and NRO Demat account.