Mammals are ectotherms, regulating their own body temperature, Arctic animals tend to be large because that minimizes the ratio of surface area to total body mass, thus making sure heat loss through the skin, while breathing, etc., doesn't exceed the ability of the internal "furnace" to pump enough warmed blood to keep the whole system going. (The heat generated by moving the muscles is also important: some whales, with thick coats of blubber to insulate them, can actually burn internally if killed by whalers and left floating without being cut open. )
That ratio obviously gets worse, from a thermal point of view, as the mammal gets smaller. One adaptation is increased heartbeat. Your heart might beat 70-100 times a minute, while a mouse's pulse starts around 300 and can exceed 800 when excited. But there's a limit to the point at which this works: we will never see a mouse the size of an ant.
What we do see, though, is pretty amazing. The size limit of mammals has been defined downward several times by discoveries in the field. The smallest rodent, the pygmy jerboa, weighs only about 3 grams. (The queen of the African wandering ant, swollen with eggs, can weigh 10g, so we have to walk back a little the "no mammals the size of ants," even though a worker ant weighs a few milligrams - 90, in the largest species.)
The jerboa isn't at the bottom of the scale, though, Kitti's hognosed bat and the Etruscan shrew are approximately tied around two grams. (The shrew's heartbeat? A modest 1,500 per minute.) It is unknown whether we might find any still smaller mammals. By definition, they won't be easy to spot.
IUCN entries for the smallest mammals:
Kitt's hog-nosed bat