filtering-results.js

computed: {
  countriesResults() {
     // Filter results
     return this.countries.filter((country) => {
       return country.name.toLowerCase()
       .indexOf(this.search.toLowerCase()) > -1;
     });
  }
}

render-filtered-results.html

<ul>
    <li v-for="country in countriesResults" :key="country.id">
      {{ country.name }}
    </li>
</ul>