Flutter firebase exception not being caught by try-catch block

I am trying to implement simple login using flutter and firebase(just started learning). The user-not-found error is not being caught by the try-catch block in the code.

Code in Question

TextButton(
  onPressed: () async {
    final email = _eMail.text;
    final password = _passWord.text;
    try {
      final userCredential =
          FirebaseAuth.instance.signInWithEmailAndPassword(
        email: email,
        password: password,
      );
      print(" UC = \n");
      print(userCredential);
    } catch (e) {
      print("Error Code: ");
      print(e.runtimeType);
      print(e);
    }
  },
  child: Text("Login"),
),

Output(in the debugger):

I/flutter (10795):  UC = 
I/FirebaseAuth(10795): Logging in as nvnv@gmail.com with empty reCAPTCHA token
I/flutter (10795): Instance of 'Future<UserCredential>'
W/System  (10795): Ignoring header X-Firebase-Locale because its value was null.
E/RecaptchaCallWrapper(10795): Initial task failed for action RecaptchaAction(action=signInWithPassword)with exception - There is no user record corresponding to this identifier. The user may have been deleted.
E/flutter (10795): [ERROR:flutter/runtime/dart_vm_initializer.cc(41)] Unhandled Exception: [firebase_auth/user-not-found] There is no user record corresponding to this identifier. The user may have been deleted.
E/flutter (10795): #0      FirebaseAuthHostApi.signInWithEmailAndPassword (package:firebase_auth_platform_interface/src/pigeon/messages.pigeon.dart:1158:7)
E/flutter (10795): <asynchronous suspension>
E/flutter (10795): #1      MethodChannelFirebaseAuth.signInWithEmailAndPassword (package:firebase_auth_platform_interface/src/method_channel/method_channel_firebase_auth.dart:343:22)
E/flutter (10795): <asynchronous suspension>
E/flutter (10795): #2      FirebaseAuth.signInWithEmailAndPassword (package:firebase_auth/src/firebase_auth.dart:590:9)
E/flutter (10795): <asynchronous suspension>
E/flutter (10795):

From what I understand, if the error was caught, the print(" UC = \n"); statement should not have been executed.

Please help me solve this problem

As DanR commented, the signInWithEmailAndPassword method is asynchronous, so you have to use await to wait for it to complete:

//                      👇
final userCredential = await FirebaseAuth.instance.signInWithEmailAndPassword(
  email: email,
  password: password,
);