Edit in JSFiddle

<div class="dropdown dropdown-scroll" ng-app="app">
    <button class="btn btn-default dropdown-toggle" type="button" id="dropdownMenu1" data-toggle="dropdown">Select <span class="caret"></span>
    </button>
    <ul class="dropdown-menu" role="menu" aria-labelledby="dropdownMenu1" ng-controller="ListCtrl">
        <li role="presentation">
            <div class="input-group input-group-sm search-control"> <span class="input-group-addon">
                    <span class="glyphicon glyphicon-search"></span>
</span>
                <input type="text" class="form-control" placeholder="Query" ng-model="query"></input>
            </div>
        </li>
        <li role="presentation" ng-repeat='item in items | filter:query'> <a href="#"> {{item.name}} </a>
        </li>
    </ul>
</div>
// Angular

var phonecatApp = angular.module('app', []);

phonecatApp.controller('ListCtrl', function ($scope) {
    $scope.items = [{
        'name': 'Item 1'
    }, {
        'name': 'Item 2'
    }, {
        'name': 'Account 3'
    }, {
        'name': 'Account 4'
    }, {
        'name': 'Item 5'
    }, {
        'name': 'Item 6'
    }, {
        'name': 'User 7'
    }, {
        'name': 'User 8'
    }, {
        'name': 'Item 9'
    }, {
        'name': 'Item 10'
    }, {
        'name': 'Item 11'
    }, {
        'name': 'Item 12'
    }, {
        'name': 'Item 13'
    }, {
        'name': 'Item 14'
    }, {
        'name': 'User 15'
    }, {
        'name': 'User 16'
    }, {
        'name': 'Person 17'
    }, {
        'name': 'Person 18'
    }, {
        'name': 'Person 19'
    }, {
        'name': 'Item 20'
    }, ];
});

// jQuery
$('.dropdown-menu').find('input').click(function (e) {
    e.stopPropagation();
});
.dropdown.dropdown-scroll .dropdown-menu {
    max-height: 200px;
    width: 60px;
    overflow: auto;
}
.search-control {
    padding: 5px 10px;
}

External resources loaded into this fiddle: