Create a custom excerpt from the first paragraph of the content.

wp_first_paragraph_excerpt.php

<?php
/**
 *  Create a custom excerpt string from the first paragraph of the content.
 *
 *  @param   integer  $id       The id of the post
 *  @return  string   $excerpt  The excerpt string
 */
function wp_first_paragraph_excerpt( $id=null ) {
    // Set $id to the current post by default
    if( !$id ) {
        global $post;
        $id = get_the_id();
    }

    // Get the post content
    $content = get_post_field( 'post_content', $id );
    $content = apply_filters( 'the_content', strip_shortcodes( $content ) );

    // Remove all tags, except paragraphs
    $excerpt = strip_tags( $content, '<p></p>' );

    // Remove empty paragraph tags
    $excerpt = force_balance_tags( $excerpt );
    $excerpt = preg_replace( '#<p>\s*+(<br\s*/*>)?\s*</p>#i', '', $excerpt );
    $excerpt = preg_replace( '~\s?<p>(\s|&nbsp;)+</p>\s?~', '', $excerpt );

    // Get the first paragraph
    $excerpt = substr( $excerpt, 0, strpos( $excerpt, '</p>' ) + 4 );

    // Remove remaining paragraph tags
    $excerpt = strip_tags( $excerpt );

    return $excerpt;
}