A working example of a horizontal ScrollView with some example click events.

doggos.jsx

import React, { Component } from 'react';
import { StyleSheet, Text, View, Button, ScrollView } from 'react-native';

class DoggoListButton extends Component
{
  handlePressButton()
  {
    //This is called when you press this doggo button
    //it just prints out the name of this doggo and its number that was passed :)
    alert("hello i am " + this.props.doggo + ", my number is " + this.props.num + " and I am the bestest boyo in the whole wide WORLD");
  }

  render()
  {
    //The doggo button is just a button but with two things: a title of this doggo
    //and an onPress event which calls the function above called handlePressButton ^ ^ ^
    return <>
      <View style={{ flex: 1, marginRight: 10 }}>
        <Button onPress={this.handlePressButton.bind(this)} title={this.props.doggo}/>
      </View>
    </>;
  }
}

class NavigationBar extends Component
{
  render()
  {
    //The list of buttons we're gonna make for this navigation bar
    let items = [];

    //An array of the BEST hecking doggos in the whole WORLD
    let specialDoggos = ["copper", "pickle", "gimms", "barkley"];

    //A function that will return a random woofer name for us
    let randomDoggo = x => specialDoggos[Math.floor(Math.random() * specialDoggos.length)];

    for(let i = 0; i < 20; i++)
    {
      //Build button with a random doggo name and the number of this doggo
      let item = <DoggoListButton key={i} num={i} doggo={randomDoggo()} />

      //Add it to the list of doggo buttons :) 
      items.push(item);
    }

    //This just makes a scrolley scrolley horizontal list of those doggo
    //buttons we just made
    return (
        <View>
        <ScrollView style={{ paddingTop: 10 }} horizontal={true}>
            {items}
          </ScrollView>
        </View>
    );
  }
}