Like @DavidArno has correctly stated in his comment

And now you have a global method with static state, which is not a good place to be.

this isn't the best way to do this. By having a static dictionary this just can't be tested correctly, because the dictionary keeps its content between tests which is a no go.

A better way would be to create an interface having the GetAvatar() method which is implemented by a simple class. The interface should then be constructor injected and stored in a private field.

public interface IAvatarService
{
    string GetAvatar(System.Security.Principal.IIdentity user);
}  

public class AvatarService : IAvatarService
{
    private ConcurrentDictionary<string, string> avatarDictionary = new ConcurrentDictionary<string, string>();
    public string GetAvatar(System.Security.Principal.IIdentity user)
    {

        if (user == null) { throw new ArgumentNullException("user"); }  
    
        string userId = user.GetUserId();
        string avatar;
        if (avatarDictionary.TryGetValue(userId, out avatar))  
        {
            return avatar;
        }

        avatar = user.GetAvatar() // call of the extension method

        avatarDictionary.AddOrUpdate(userId, avatar, (key, oldValue) => avatar);
        
        return avatar;
    }
}

having a expire mechanism would still be needed and should be implemented.

public static string GetAvatar(this System.Security.Principal.IIdentity user)
{
    var service = SmObjectFactory.Container.GetInstance<IApplicationUserManager>();
    return service.FindByIdAsync(int.Parse(user.GetUserId())).Result.Avatar;
}  

Do you know that you can access this static method not only as a extension method ? Assume the containing class is called UserExtensions one could execute this method using UserExtensions.GetAvatar(null) and if the creation of the service would take some time to be created it would nevertheless throw an ArgumentNullException. So adding a guard condition to the method like

if (user == null) { throw new ArgumentNullException("user"); }  

would be a good idea.

Otherwise the method is looking good.

By using a private static ConcurrentDictionary<string, string> you could add caching to this like so

private static ConcurrentDictionary<string, string> avatarDictionary = new ConcurrentDictionary<string, string>();
public static string GetAvatar(this System.Security.Principal.IIdentity user)
{

    if (user == null) { throw new ArgumentNullException("user"); }  

    string userId = user.GetUserId();
    string avatar;
    if (avatarDictionary.TryGetValue(userId, out avatar))  
    {
        return avatar;
    }

    var service = SmObjectFactory.Container.GetInstance<IApplicationUserManager>();
    avatar = service.FindByIdAsync(int.Parse(user.GetUserId())).Result.Avatar;

    avatarDictionary.AddOrUpdate(userId, avatar, (key, oldValue) => avatar);
    
    return avatar;
}

public static string GetAvatar(this System.Security.Principal.IIdentity user)
{
    var service = SmObjectFactory.Container.GetInstance<IApplicationUserManager>();
    return service.FindByIdAsync(int.Parse(user.GetUserId())).Result.Avatar;
}  

Do you know that you can access this static method not only as a extension method ? Assume the containing class is called UserExtensions one could execute this method using UserExtensions.GetAvatar(null) and if the creation of the service would take some time to be created it would nevertheless throw an ArgumentNullException. So adding a guard condition to the method like

if (user == null) { throw new ArgumentNullException("user"); }  

would be a good idea.

Otherwise the method is looking good.

By using a private static ConcurrentDictionary<string, string> you could add caching to this like so

private static ConcurrentDictionary<string, string> avatarDictionary = new ConcurrentDictionary<string, string>();
public static string GetAvatar(this System.Security.Principal.IIdentity user)
{

    if (user == null) { throw new ArgumentNullException("user"); }  

    string userId = user.GetUserId();
    string avatar;
    if (avatarDictionary.TryGetValue(userId, out avatar))  
    {
        return avatar;
    }

    var service = SmObjectFactory.Container.GetInstance<IApplicationUserManager>();
    avatar = service.FindByIdAsync(int.Parse(user.GetUserId())).Result.Avatar;

    avatarDictionary.AddOrUpdate(userId, avatar, (key, oldValue) => avatar);
    
    return avatar;
}

Based on @mjolka's comment

Using a ConcurrentDictionary as a cache in this manner could be problematic, depending on the number of users and the size of the avatars. As they say, "A cache without an expiration policy is another name for a memory leak."

you should add some mechanism to let the entries of the dictionary expire.