The trick here is to use event.stopPropagation() which will just stop event bubbling to HTML elements that are higher in the hierarchy.

The delete handler that stops propagation.

  const deleteItem = (e, index) => {
    e.stopPropagation();
    e.preventDefault();
    const updated = [...items];
    updated.splice(index, 1);
    setItems(updated);
  };

The whole code with a button that deletes an item from a dropdown.

import React from "react";
import "antd/dist/antd.css";
import "./index.css";
import { PlusOutlined } from "@ant-design/icons";
import { Divider, Input, Select, Space, Typography, Button } from "antd";
import { useState } from "react";
const { Option } = Select;
let index = 0;

const App = () => {
  const [items, setItems] = useState(["jack", "lucy"]);
  const [name, setName] = useState("");

  const onNameChange = (event) => {
    setName(event.target.value);
  };

  const addItem = (e) => {
    e.preventDefault();
    setItems([...items, name || `New item ${index++}`]);
    setName("");
  };

  const deleteItem = (e, index) => {
    e.stopPropagation();
    e.preventDefault();
    const updated = [...items];
    updated.splice(index, 1);
    setItems(updated);
  };

  return (
    <Select
      style={{
        width: 300
      }}
      placeholder="custom dropdown render"
      dropdownRender={(menu) => (
        <>
          {menu}
          <Divider
            style={{
              margin: "8px 0"
            }}
          />
          <Space
            align="center"
            style={{
              padding: "0 8px 4px"
            }}
          >
            <Input
              placeholder="Please enter item"
              value={name}
              onChange={onNameChange}
            />
            <Typography.Link
              onClick={addItem}
              style={{
                whiteSpace: "nowrap"
              }}
            >
              <PlusOutlined /> Add item
            </Typography.Link>
          </Space>
        </>
      )}
    >
      {items.map((item, index) => (
        <Option key={item}>
          <div
            style={{
              display: "flex",
              justifyContent: "space-between",
              alignItems: "center"
            }}
          >
            <div>{item}</div>
            <Button onClick={(e) => deleteItem(e, index)} danger size="small">
              Delete
            </Button>
          </div>
        </Option>
      ))}
    </Select>
  );
};

export default App;