It would be for 4 ppl max and would need to be well insulated (I live in a very cold climate.)

Many DIY sites offer information about building an outdoor sauna. In addition, your county agency should have the ordinances for building codes and regulations. Many require permits for building or adding.

I have no electricity (and have no desire to get any). I live out in the mountains of a tropical country.

You will not need electricity for this method.
1. Decide where you want your sauna.
Make sure you have available water close by.

You will need a large supply of rocks (large rocks, and med.sized rocks).
Method most natural: North American Native Sweat Lodge.
Find 8 branches about 6 feet long and remove all leaves, twigs and bark. And one small tree about 5 inches around and 8 ft long.
Lay the branches out in a pie shape so that they all meet in the center of the circle. Tie these together with bark strips or rope.
Dig a deep hole about 3 ft deep in the center of you sauna area. Place the small tree , devoid of branches, in the hole fill with soil ( mixture of clay 75%,dirt 25%) and damper it down, this becomes your center pole. Wet the soil around the pole, pack firmly and let set for two days.
Re-dampen and add more soil, damper down again. Add 10% more clay to soil, and enough water to make a potters clay material. Build this up around the bottom of the pole.
Collect your rocks.
When the pole is firmly set drape you pie shaped branches over the top of it, making sure it is centered. Stake the branches down to the ground or stap them together.
Take green boughs from forest and weave them between the branches of your frame. Weave around first then weave verticaly until you have a full tight frame but have left room to enter at one side. Leave some space ( loose weave) at the top to allow for smoke ventilation. Inside and off to the side of the pole constuct a fire pit out of the stones. This does not have to be very big. Never build a large fire in this hut. Your main purpose is to heat the stones. Put a pile of stones and wood at the outer wall for use in sauna.
Cover the outside with natural materials like canvas tarp (hides) or large leaves to make it keep the moisture and heat in. Make a door covering to close off the door when in use. This can be a simple frame done the same way as your hut that you move aside to get in and out. Make it larger than the entrance.
To Use, pack 3 large containers of water and place inside your hut.
Put in a sitting mat or stool.
Build a fire, get your self ready. When fire is to hot coals start to dribble water over the stones to create steam. Sit your self near the stones. Do not put water directly on the fire. If you want more steam faster, then add hot rocks to a container of water, removing them as they cool and replacing with more hot rock. This is where your extra pile of stones come into work,pile them up around the fire. Add more wood to fire as it burns down to coals. You can add any frangrance you want to the water to create a aromatherpy bath.
Sit and Enjoy.
This hut can be used for many seasons just by replacing the outter cover.