The fight between Bando and Nana is a rather curious and unique battle in Elfen Lied, for it's the only fight where neither of the combatants suffer permanent injury or death, and both part on, if not friendly, at least somewhat cordial terms.
It all began when Bando, in his bid to clean up the beach of all waste, encountered a giant pod the currents dragged to the shore. From it, emerged Nana, lost and confused.
It turns out Nana was made unconscious by her Papa, Kurama before being sent away from the Facility, where Director Kakuzawa demanded her execution. Bando, however, had no knowledge, nor interest, in the girl's plight, and just pointed his gun at her as soon as he noticed her horns.
Incensed by his demands of knowing where Lucy was, Nana simply answered that he should leave her alone, because her Papa ordered her not to hurt anyone, but to not jump to the conclusion that she was harmless.
Contrary to those warnings, Bando proceeded to open fire on her, promptly knocking her off-balance. To Nana's shock, Bando was already well prepared to fight a Diclonius like her, his Desert Eagle loaded with heavy 50. Caliber tungsten bullets.
Despite this attack, however, Nana hesitated in attacking Bando. thinking back to her Papa's words. This, however, did nothing but embolden the violent man, who continued attacking and taunting her. His cruelty and taunts began grating on Nana's nerves, slowly but surely causing the mental defences she'd set up to crumble, exposing her to a cold, harsh realization: her entire species were essentially enemies of humanity, kept alive only to experiment on them.
Despite desperately trying to deny it, Bando's words cut deep into her. And then, something inside her broke.
To Bando's surprise, the girl stood up, mumbling to herself about carving a place in the world, before turning to face him. Her eyes now looked dead, much like Lucy's own, as she flared up her vectors, the intent to kill evident.
Bando quickly resumed fire, however he was unable to land an incapacitating blow on her, as the bullets he aimed at her limbs only grazed her artificial limbs.
Bando had made a critical mistake. Assuming Nana's range was the same as Lucy. This, coupled with his complete ignorance of the different vector ranged from individual Diclonii, meant he was more unprepared to face Nana that he had imagined.
As she approached, Bando thought that he could just shoot her at a certain distance, only for her to literally shoot one of her limbs directly at Bando.
The man was caught completely off-guard by her move, and was knocked down as a result. His attempt to reach for the gun he dropped quickly foiled as Nana soon overpowered him, putting him in a position not unlike that of his first duel with Lucy.
As Nana towered over him, she repeated the same callous taunts he threw at her, ordering him to beg for his life, as he tried to make her.
Bando, however, was nowhere near as helpless as he had been in his past battle with Lucy. In a sudden move, he pulled out a second gun from his jacket and fired it directly at Nana, hoping to take her life with it.
However, the limitations of his prosthetic limb ensured he would not be able to land a lethal blow, the arm disassembling as soon as the force of the recoil hit it. The bullet itself missed Nana's skull for a few inches, only grazing her head.
This proved enough to shake Nana out of her despair-induced rage and reduce her to a crying heap on the ground. The concept of a crybaby Diclonius was enough to put Bando off, and the two quickly realized a common hatred of Lucy, as well as a desire to end her threat. The battle was called off when Nana, who could sense others like her, agreed to do so with Lucy and report back to Bando. The conclusion, as with so many moments with these two characters, came in the form of slapstick when Nana offered Bando a hand up, only for his damaged prosthetic hand to come apart, with him falling back to the sand amidst accusations Nana had planned this.
Bando's relationship with Nana stayed firmly in the middle of the dial, so to speak, neither connecting with her the way he did with Mayu, nor gaining the mutual intense hatred he shared with Lucy. In another anomaly, the anime version of this battle differed very little from the manga version (Users please check that last statement - Gojirob)