WordPress strips out tags in `wp_trim_words()`, which is called by `get_the_excerpt()`; so we have to filter 'wp_trim_words', basically copying that function with one change: replace `wp_strip_all_tags()` with `strip_tags()`. We don't want other functions that run `wp_trim_words` to be modified, so we add our filter while `get_the_excerpt()` is …

allow-links-in-excerpts.php

// Allow links in excerpts
function sg_trim_words( $text, $num_words, $more, $original_text ) {
	$text = strip_tags( $original_text, '<a>' );
	// @See wp_trim_words in wp-includes/formatting.php
	if ( strpos( _x( 'words', 'Word count type. Do not translate!' ), 'characters' ) === 0 && preg_match( '/^utf\-?8$/i', get_option( 'blog_charset' ) ) ) {
		$text = trim( preg_replace( "/[\n\r\t ]+/", ' ', $text ), ' ' );
		preg_match_all( '/./u', $text, $words_array );
		$words_array = array_slice( $words_array[0], 0, $num_words + 1 );
		$sep = '';
	} else {
		$words_array = preg_split( "/[\n\r\t ]+/", $text, $num_words + 1, PREG_SPLIT_NO_EMPTY );
		$sep = ' ';
	}
	if ( count( $words_array ) > $num_words ) {
		array_pop( $words_array );
		$text = implode( $sep, $words_array );
		$text = $text . $more;
	} else {
		$text = implode( $sep, $words_array );
	}
	// Remove self so we don't affect other functions that use wp_trim_words
	remove_filter( 'wp_trim_words', 'sg_trim_words' );
	return $text;
}
// Be sneaky: add our wp_trim_words filter during excerpt_more filter, which is called immediately prior
function sg_add_trim_words_filter( $excerpt_length ) {
	add_filter( 'wp_trim_words', 'sg_trim_words', 10, 4 );
	return $excerpt_length;
}
add_filter( 'excerpt_more', 'sg_add_trim_words_filter', 1 );