We have Chihuahuas as a family member for almost 30 years and never had a problem with house training.
Just be patient with your pup and be consistent in training, Chihuahuas are very good at manipulating their people and get things their way, so just do not give up.
Chihuahuas are intelligent dogs, you can learn your little dog to do his business indoors on a litter box or outdoors on command.
Or even both like we did, it will be beneficial if your Chihuahua can go potty on a litter box when you are not feeling well and cannot go out for a walk.
When we bring a new puppy home we place a puppy trainer near the crate and a dog bed all in the same room.
It makes it easier for your pup to get used to the new environment when he has everything he needs in one room.
Pick one command you like, something like "go potty", "do your business" or "Get busy" and use this every time you pup needs to do his business.
Watch you Chihuahua pup closely and if you see your pup needs to eliminate, take him to the puppy trainer and use the command you like to use.
Repeat the command and as soon as your pup eliminates you need to praise him and reward him with something he wants.
We always give a small puppy kibble, it does not take long and your dog will know this ritual.
Remember a young puppy needs to go after a nap, shortly after eating, often while playing...
It is your responsibility to watch your dog, do not blame your pup when it goes wrong.
Accidents will happen while potty training a pup, just ignore and clean it up.
When you cannot closely watch the pup while doing home chores, cooking or you need to go out place your puppy in a crate.
Most puppies will not soil their bed, just make sure the crate is not to big.
After a few days or a week your pup will feel comfortable enough to explore other rooms and you can move the puppy trainer closer to the door.
When your Chihuahua pup has grown big enough to step into a litter box you can replace the trainer, but keep on using the command.
Besides doing indoor training we do take our pups outdoors on a leash for a walk and to learn to feel comfortable outside of the house, to meet other dogs and people.
Although a Chihuahua is a tiny dog breed, he likes to do what all dogs do and while sniffing here and there, as soon as you see your dog wants to relieve himself give the command.
It is possible you will walk with your pup doing nothing at all and when you come back home he will run to the litter box.
Do not give up, just try again the next time you go out for a walk.
When your pup does start to do his business outdoors, praise him like you did indoors and give him a little treat.
It will not take long before your dog gives you a signal that he needs to go out, you do not have to give a treat anymore, a simple "good boy" will be enough.
At that point you can decide to remove the litter box, or maybe you still like to use it when you do not feel well or do not like to go out late at night.
Whatever your decision about potty training is, your Chihuahua although tiny is a dog, with dog needs and he likes to go out for a walk every day if possible.
Our Chihuahuas go out at least 4 times a day they do not care if it is raining and love to play in the snow, they love to be outdoors.