I don't know of any who did, but Sparta certainly didn't. There is not a single shred of evidence that the Spartans worked to improve their physical strength or combat ability.

It is very important not to project onto Sparta what we expect a society of warriors to be. As far as we can tell from contemporary sources, the Spartan upbringing was focused entirely on obedience and endurance; the military training of adult Spartans, meanwhile, consisted only of formation drill. That was pretty much it. The perfect Spartan was an automaton, unflinchingly obedient, indifferent to cold, heat, hunger, thirst or fatigue, deferential to his elders and more afraid of disgrace than death. He was not a hulking badass action hero. No source utters a word about Spartans training in the use of weapons or working on their sheer muscle strength.

In fact, the Greeks in general tended to laugh at those who trained in weapon proficiency, and professional athletes (whose diet and training focused on bulking up and becoming as strong as possible) were deemed unfit for war. The only form of sport that was considered useful preparation for battle was wrestling, and all we know about wrestling at Sparta is Plutarch's claim that it was banned. In terms of importance, the mere physical aspects of a good warrior were considered a very distant second to the mental aspects I just listed.

Because of Sparta's history of relative stability and military success, people in later periods of Antiquity became fascinated with them, wondering what it was that made them so special. Sparta itself had by this time become a quaint backwater in a poor part of the Roman Empire; they welcomed the tourism of fascinated outsiders, and played up their own weirdness for their sake. A lot of stories about Sparta originate in this time. The story of them killing babies because they seemed unhealthy is one of them. We really shouldn't put too much stock by these tall tales.

There's a great example of the way in which the Spartans themselves distorted their own image for the sake of Roman tourists. In the 4th century BC, when Sparta was still relevant, Xenophon wrote about a ritual that took place at the temple of Artemis Orthia. A table was set up with a pile of cheeses on it, guarded by young men with whips. Boys were encouraged to try and steal as many cheeses as they could without being whipped. This was supposed to train their cunning and agility, and, if necessary, their endurance of pain. Now, we hear of this ritual again from Cicero, some 300 years later - but by his time it had taken a very dark turn. There were no more cheeses. Instead, youths were simply lined up and whipped until they collapsed. The contest was about who could last the longest. This was how the Spartans proved to the Romans that they were the hardest, most ruthless people of all. An entire theatre was built around the temple to allow tourists to watch the spectacle.

It's useful to keep this example in mind when you read anything about Sparta that is derived from a late source.