Can a barbarian use a spell like ability while raging?
No, because raging disallows actions that require concentration.
It's not explicit, but close reading of the rules shows that spell-like abilities require concentration (d20pfsrd, emaphsis mine):
Spell-Like Abilities (Sp)
Usually, a spell-like ability works just like the spell of that name. A spell-like ability has no verbal, somatic, or material component, nor does it require a focus. The user activates it mentally. Armor never affects a spell-like ability's use, even if the ability resembles an arcane spell with a somatic component.
A spell-like ability has a casting time of 1 standard action unless noted otherwise in the ability or spell description. In all other ways, a spell-like ability functions just like a spell.
Spells require concentration, and therefore spell-like abilities require concentration unless a rule specifically says otherwise. Nothing on that page mentions anything of the sort. This is in contrast to supernatural and extraordinary abilities, which "cannot be disrupted in combat".
This conclusion is reflected in the tactics of a NPC in the Hell's Rebels adventure path.
[character] avoids using rage on themselves since they can't use [spell-like ability] while under its effects. (In Hell's Bright Shadow, p. 49)
A character cannot use actions that require concentration during a Barbarian's Rage:
While in rage, a barbarian cannot use any Charisma-, Dexterity-, or Intelligence-based skills (except Acrobatics, Fly, Intimidate, and Ride) or any ability that requires patience or concentration.
So, for spell-like abilities:
Usually, a spell-like ability works just like the spell of that name.
A spell-like ability has no verbal, somatic, or material component, nor does it require a focus.
From the corerule book, on the chapter Magic, also found here:
To cast a spell, you must be able to speak (if the spell has a verbal component), gesture (if it has a somatic component), and manipulate the material components or focus (if any). Additionally, you must concentrate to cast a spell.
You must concentrate to cast a spell. If you can't concentrate, you can't cast a spell. If you start casting a spell but something interferes with your concentration, you must make a concentration check or lose the spell.
More references on how spell-like abilities work:
Page 183, table 8-2, Spell-like Abilities trigger AoO.
Page 554, spell-like abilities function exactly like spells of the same name.
Page 221, they do not require verbal, somatic or material components. Are not affected by armor. They require a standard action and are considered to have a "casting time".
A spell-like ability has a casting time of 1 standard action unless noted otherwise in the ability or spell description
The user activates it mentally. Armor never affects a spell-like ability's use, even if the ability resembles an arcane spell with a somatic component.
At 4th level, the bloodrager gains the ability to cast spells even while bloodraging. He can also cast these spells defensively and can make concentration checks for these spells while bloodraging. While bloodraging, he can cast and concentrate on only his bloodrager spells (see below); spells from other classes cannot be cast during this state.
While in rage, a barbarian cannot use [...] any ability that requires patience or concentration.
Since spell-like abilities require concentration checks (e.g. if you get hit while trying to cast one), they cannot be used in a rage.
Supernatural abilities, on the other hand, cannot be disrupted and do not involve concentration checks, so those can be used in a rage.