Can a Warlock use his Eldritch blast when paralyzed?

Question

A warlock's eldritch blast is a spell-like ability:

Eldritch Blast (Sp): The first ability a warlock learns is eldritch blast. A warlock attacks his foes with eldritch power, using baleful magical energy to deal damage and sometimes impart other debilitating effects. An eldritch blast is a ray with a range of 60 feet. It is a ranged touch attack that affects a single target, allowing no saving throw. An eldritch blast deals 1d6 points of damage at 1st level and increases in power as the warlock rises in level. An eldritch blast is the equivalent of a spell whose level is equal to one-half the warlock’s class level (round down), with a minimum spell level of 1st and a maximum of 9th when a warlock reaches 18th level or higher.

Spell like Abilities

A spell-like ability has no verbal, somatic, or material component, nor does it require a focus or have an XP cost. 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 spelllike ability has a casting time of 1 standard action unless noted otherwise in the ability or spell description. In all other ways, a spelllike ability functions just like a spell.

Edit: From Complete Arcane p7 (under Weapons and Armor, not Invocations, go figure) (as pointed out by ZanathKariashi below)

Because the somatic components required for warlock invocations are relatively simple, a warlock can use any of his invocations while wearing light armor without incurring the normal arcane spell failure chance. However, like arcane spellcasters, a warlock wearing medium or heavy armor or using a shield incurs a chance of arcane spell failure (all invocations, including eldritch blast, have a somatic component).

Can a warlock fire an eldritch blast at an opponent when paralyzed? This attack is normally described in our game as the warlock pointing at his target and the ray firing from his finger. Is that a requirement or just showmanship?

Answer 1

No.

It specifically mentions that warlock eldritch blasts and invocations are subject to arcane spell failure because unlike most spell-like abilities they have somatic and/or verbal components, albeit simple ones that can be cast while wearing light armor (or medium with Battle Caster feat).

So in that situation, no they couldn't. And since spell-like abilities are in all other circumstances non-somatic/verbal, there's no Still/Silent Spell-Like Ability feat (unlike Empower/Quicken/Maximize). So you'll never be capable of casting them in that situation.

From Complete Arcane, page 7:

All invocations, including eldritch blast, have a somatic component.

[...]

Unlike other spell-like abilities, invocations are subject to arcane spell failure chance.

Answer 2

As the previous answer gave, "No" is the correct answer. However, it would be very reasonable for you to talk to your DM about granting a metamagic-like feat similar to empower spell-like ability that worked for still spell-like ability. That feat will never exist as an official feat because warlock invocations are (to my knowledge) the only spell-like abilities requiring a somatic component.

Also, despite their descriptions, none of the "word" invocations (baleful utterance, word of changing, et al.) actually require a verbal component, either. All invocations have a somatic and only a somatic component (with the rare material component, such as animate dead).