Intenta de esta forma
Código PHP:
$servidor = "";
$nombreBD = "";
$usuario = "";
$contrasena = "";
$conexion = new mysqli($servidor,$usuario,$contrasena,$nombreBD);
if($conexion->connect_error){
die("Error en la conexion : ".$conexion->connect_errno.
"-".$conexion->connect_error);
}
echo "<select name='combopaises'>";
$query = "SELECT short_name FROM country_t";
$stmt = $conexion->prepare($query);
$stmt->execute();
$stmt->bind_result($exist);
while($stmt->fetch()){
echo "<option value='$exist'>$exist</option>";
}
echo "</select>";
$stmt->close();
$conexion->close();