Simple drop-down like layout.

main.dart

import 'package:flutter/material.dart';

final Color darkBlue = Color.fromARGB(255, 18, 32, 47);

void main() {
  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      theme: ThemeData.dark().copyWith(scaffoldBackgroundColor: darkBlue),
      debugShowCheckedModeBanner: false,
      home: Scaffold(
        body: Center(
          child: MyWidget(),
        ),
      ),
    );
  }
}

class MyWidget extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return Container(
      width: 160,
      height: 40,
      decoration: BoxDecoration(
          borderRadius: BorderRadius.all(Radius.circular(100)),
          color: Colors.white,
      ),
      
      child: Stack(
        children: [
          Positioned.fill(
            child:  Center(
              child: Text(
                "YYYY",
                style: TextStyle(color: Colors.black, fontSize: 20)
              ),
            ),
          ),
          
          Align(
            alignment: Alignment.centerRight,
            child: Padding(
              padding: EdgeInsets.only(right: 10),
              child: Icon(
                Icons.arrow_drop_down,
                color: Colors.black,
                size: 32,
              )
            ),
          ),
          
        ]
      
      )
      
    );
  }
}