The useImperativeHandle hook is the one you need here, And forwardRef, useRef hooks. I have created a new TextInput called ControlledTextInput to pass a react ref to it and expose the blur method outside (App component level). Note that there are two refs one is coming as a prop(to which we bind the functions) and the one inside ControlledTextInput(which is the ref to the actual text input)

import React, { useImperativeHandle, forwardRef, useRef } from "react";
import { StyleSheet, Text, View, TextInput, Button } from "react-native";

const ControlledTextInput = forwardRef((props, ref) => {
  const internalInputRef = useRef();
  useImperativeHandle(ref, () => ({
    blur: () => {
      internalInputRef.current.blur();
    }
  }));

  return (
    <TextInput
      placeholder={"Type something"}
      style={styles.textInputStyle}
      ref={internalInputRef}
    />
  );
});

const App = () => {
  const inputRef = useRef(null);

  return (
    <View style={styles.app}>
      <View style={styles.header}>
        <Text style={styles.title}>Focus lose when button click</Text>
      </View>
      <ControlledTextInput ref={inputRef} />
      <Button title="Click ME" onPress={() => inputRef.current.blur()} />
    </View>
  );
};

const styles = StyleSheet.create({
  app: {
    marginHorizontal: "auto",
    maxWidth: 500
  },
  header: {
    padding: 20
  },
  title: {
    fontWeight: "bold",
    fontSize: "1.5rem",
    marginVertical: "1em",
    textAlign: "center"
  },
  textInputStyle: {
    border: "2px solid black",
    maxWidth: 500,
    height: 50,
    textAlign: "center",
    fontSize: 20
  }
});

export default App;

React Native Codesandox: