Curious about feltmaking? Know a playful cat? Here's a minimal-investment introduction to needle-felting (easier precise control) and wet-felting (strong material faster). The result is a cat toy that can take a licking (and swatting, and biting, and clawing).

No knitting, crocheting, sewing, or gluing.
Fast and simple enough for a class project (does involve SHARP and HOT objects).
This product was tested on animals. They liked it.

Step 1: You Need:

Foam ball, ~1”/25mm dia. Many foam types work; should be easily pierced by felting needles.
Wool-fiber roving, ~1/4 oz, cut 5”/125mm long. If roving is dyed, be sure dye is nontoxic and impervious to kitty drool. Store-bought is no guarantee. Undyed roving avoids the whole issue.
Scissors to cut wool fiber
Soap and water. Liquid hand-soaps work well; dyers' high-surfactants like Synthrapol^TM are excellent.
Needle-felting tool (1 to 7 needles).
Needle-felting backstop: an extra chunk of sturdy foam or a needle-felting mat (>1”/25mm in all dimensions). This protects the specially barbed felting needles, the furniture, and any nearby living flesh. Felting needles crave blood, especially from careful, coordinated people who never hurt themselves with anything else.
Microwave oven
Clothes dryer (last step – students may do this part at home).

Step 2: Set Up the Body and Tail

Spread out the roving from one end, forming a funnel in the middle. For best results, the funnel wall should be fairly uniform in thickness. If splits happen, gently pull fibers from the sides of the split to cover the gap.
Put the ball into the funnel.
Pinch the funnel closed. Where you're pinching will be the nose end.
Pinch the roving behind the ball to trap the ball. The long trailing end will be the tail.
Cut through the tail section diagonally to make it taper from the base (where you're pinching) to the far end.
Save the section you just cut off – you'll make the ears out of it later.

Step 3: Start Wet-felting the Body

Soap and mechanical “working” make the fibers tangle into felt. Heat shrinks them into a tighter, stronger tangle. Too much heat too soon can inhibit the tangling by restricting the fibers' "wiggle-room," resulting in a weaker felt - so we'll start out just mildly warm.

Add about ¼ tsp. of liquid soap (or 2-3 drops Synthrapol) to 1 cup of warm water (105-115F / 40-46C).
Pinching the roving at the nose and tail-base, dip the body into the soapy water. Wet the fibers all the way through.
Remove the body from the water, letting the tail hang free so it doesn't accidentally stick to the body. Squeeze the tail a couple of times so it doesn't run water continuously, but leave it pretty wet. At this point the water helps the fibers move past each other to start forming felt..
Pinch the nose and the base of the tail and twist them a little to anchor the fibers around the ball..
Make sure you are at least 2'/0.7m from anything that drops of water could damage. With the nose pointing upwards, wrap both hands around the body. Conform your fingers around the curve of the ball, but don't squeeze hard enough to deform the ball. Roll the body back and forth between your fingers (like making a clay snake that just ate a big meal), letting the tail hang free, for 30 seconds. You're getting the fibers to hug the ball and stick together by themselves at the nose and tail-base. You can go pretty fast if you don't mind getting a little damp.
Now take the body by the tail and roll the tail the same way (like making a “normal” clay snake). Here you can press harder right away.
Alternate rolling the body and tail as above until the nose, tail and covering of the body have the basic shape you want and start to feel like felt (takes maybe 10min). The soap will probably lather; that's OK for now.

Step 4: Form and Attach the Ears by Needle-felting

Take the triangle of roving you cut off the tail and divide it into two roughly equal triangles. Then, for each of the two triangles:

Fold it over into a mouse-ear shape, leaving roughly 3.8”/9mm length of loose fiber at the base.
Optional but recommended: If you're using a brush-like felting mat as pictured here, cover the bristles with a sacrificial scrap of netting, lightweight fabric, or paper towel so your work won't stick to the mat. Peel the work up off the mat after every 10-15 needle stabs, for the same reason.
Hold the formed ear against the backstop by the base (where the ear will attach to the head). With the needle-felting tool, stab every part of the ear EXCEPT the base until the ear keeps its shape by itself (with the 7-needle tool and merino wool, this'll take 20-40 stabs).
Attach the ear to the head: Spread the loose fibers at the base of the ear into two sections: one to go forward from the ear and one backward. Needle-felt the two sections to the body so the ear stays in place at the desired angle (maybe 30 stabs). Go ahead and stab through the surface of the foam ball; any small crumbs this creates will be contained by the felt.
Make sure the ear-base fibers are needled down all the way to the ends. It can still look fuzzy but there shouldn't be clumps of neighboring loose ends. Those might decide to grab onto each other, forming an unsightly independent tentacle or flap. (If you notice an unwanted appendage forming at any point, stop, needle it firmly to the nearest intentional surface, then resume).

Note: In my very first prototypes I tried to make the ears by wet-felting instead of needle-felting. It's possible, but much more time-consuming and difficult to control.

Step 5: Wet-felt the Assembled Form

Re-wet the mouse in the cup of soapy water.
Position the mouse in your hands for rolling again, letting the ears poke out between your pointer and middle fingers.
Roll the body for 1-3 minutes, paying special attention to the areas just in front of and in back of the ears where the new material is attached.

Step 6: Fulling (shrinking and Strengthening) the Felt

Re-wet the mouse in the soapy water and microwave it on High for 30 seconds. (Never do this with a real mouse, of course. . .)
Wait until the mouse is just cool enough to touch (e.g., when it stops steaming).
Roll the body and tail between your hands, and pinch / wiggle the ears, for 2-5 min. The felt is getting stronger, so now you can squeeze harder and twist the nose, ears and tail. You can even roll the entire body in all directions without the ears and tail sticking to the body. However, if the foam of the interior ball is crumble-prone, don't squash the body too hard.
Check the shape. If the ears, nose or tail seem to have deformed or shifted undesirably, pull them back into alignment. If they don't stay by themselves, anchor them there with the needle-felting tool.
Re-wet and microwave on High for 1 minute. Repeat the partially cooling, rolling/pinching/twisting, and checking/correcting.
Re-wet and microwave on High for 2 minutes. Repeat the partially cooling, rolling/pinching/twisting, and checking/correcting.
Now squeeze and pinch the mouse under clear running water until you can't see any more soapsuds. Pat dry.
Finish fulling in the dryer. (This step is for extra durability and may be skipped if the target audience isn't a highly active, needle-clawed Destructo Kitty). Dry for 30 minutes on High heat. (OK to dry multiple mice at the same time. Other slightly damp, lightweight, white-or-colorfast things can also share the dry-cycle - just not bulky wet sheets, towels or jeans).

Optional Extras:

Scent: Catnip leaves, oil or extract can be rubbed on, but many cats are attracted to the real-wool smell the mouse already has (most commercial craft-and-toy felts are either synthetic or processed beyond feline recognition).

Suspension: Attach sturdy twine, elastic or leather thong. You can hold the other end as-is or tie it to a doorknob, banister or dowel rod.

Decoration: If your mouse might be chewed upon, please avoid additions that might toxically dissolve, fall off and be swallowed or choked on, or fray into long stringy bits that could be partially swallowed or get stuck between teeth. If your mouse will not be chewed, go ahead, go nuts!

Repair:

If one day you find your mouse looking more like a dust bunny, you can reshape the basic features (if it's that far gone) by needlefelting. Then just dip it in warm soapy water, roll it with your hands for a few minutes, and throw it in a hot dryer for 15-30 minutes.
I have never tried putting one of these through the wash, but I'd be fairly optimistic.